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  • 1. Avgustinovich, S. V.
    et al.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solov'eva, F. I.
    On intersection problem for perfect binary codes2006In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 39, no 3, p. 317-322Article in journal (Refereed)
    Abstract [en]

    The main result is that to any even integer q in the interval 0 <= q <= 2(n+1-2) (log(n+1)), there are two perfect codes C-1 and C-2 of length n = 2(m) -1, m >= 4, such that vertical bar C-1 boolean AND C-2 vertical bar = q.

  • 2. Avgustinovich, S. V.
    et al.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    Solov'eva, F. I.
    The classification of some perfect codes2004In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 31, no 3, p. 313-318Article in journal (Refereed)
    Abstract [en]

    Perfect 1-error correcting codes C in Z(2)(n), where n = 2(m) - 1, are considered. Let [C] denote the linear span of the words of C and let the rank of C be the dimension of the vector space [C]. It is shown that if the rank of C is n - m + 2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n - m + 2, described by a Hamming code H and a set of MDS-codes D-h; h is an element of H, over an alphabet with four symbols. The case of rank n - m + 1 is much simpler: Any such code is a Vasil'ev code.

  • 3. Avgustinovich, S. V.
    et al.
    Solov'eva, F. I.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On partitions of an n-cube into nonequivalent perfect codes2007In: Problems of Information Transmission, ISSN 0032-9460, E-ISSN 1608-3253, Vol. 43, no 4, p. 310-315Article in journal (Refereed)
    Abstract [en]

    We prove that for all n = 2(k)-1, k >= 5. there exists a partition of the set of all binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with distance 3.

  • 4. Avgustinovich, S. V.
    et al.
    Solov'eva, F. I.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the structure of symmetry groups of Vasil'ev codes2005In: Problems of Information Transmission, ISSN 0032-9460, E-ISSN 1608-3253, Vol. 41, no 2, p. 105-112Article in journal (Refereed)
    Abstract [en]

    The structure of symmetry groups of Vasil'ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil'ev code of length n is always nontrivial; for codes of rank n - log(n + 1) + 1, an attainable upper bound on the order of the symmetry group is obtained.

  • 5. El-Zanati, S.
    et al.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Seelinger, G.
    Sissokho, P.
    Spence, L.
    Vanden Eynden, C.
    Partitions of the 8-Dimensional Vector Space Over GF(2)2010In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 18, no 6, p. 462-474Article in journal (Refereed)
    Abstract [en]

    Let V=V(n,q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a. partition P of V with exactly a(i) subspaces of dimension i for 1 <= i <= n, we have Sigma(n)(i=1) a(i)(q(i)-1) = q(n)-1, and we call the n-tuple (a(n), a(n-1), ..., a(1)) the type of P. In this article we identify all 8-tuples (a(8), a(7), ..., a(2), 0) that are the types of partitions of V(8,2).

  • 6. Guzeltepe, Murat
    et al.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Perfect Mannheim, Lipschitz and Hurwitz weight codes2014In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 19, no 2, p. 253-276Article in journal (Refereed)
    Abstract [en]

    The set of residue classes modulo an element pi in the rings of Gaussian integers, Lipschitz integers and Hurwitz integers, respectively, is used as alphabets to form the words of error correcting codes. An error occurs as the addition of an element in a set E to the letter in one of the positions of a word. If epsilon is a group of units in the original rings, then we obtain the Mannheim, Lipschitz and Hurwitz metrics, respectively. Some new perfect 1-error-correcting codes in these metrics are constructed. The existence of perfect 2-error-correcting codes is investigated by computer search.

  • 7.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A full rank perfect code of length 312006In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 38, no 1, p. 125-129Article in journal (Refereed)
    Abstract [en]

    A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last open case of the rank-kernel problem of Etzion and Vardy.

  • 8.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    A maximal partial spread of size 45 in PG(3,7)2001In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 22, no 3, p. 331-334Article in journal (Refereed)
    Abstract [en]

    An example of a maximal partial spread of size 45 in PG(3, 7) is given. This example show's that a conjecture of Bruen and Thas from 1976 is false. It also shows that an upper bound for the number of lines of a maximal partial spread, given by Blockhuis in 1994, cannot be improved in general.

  • 9.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A note on the symmetry group of full rank perfect binary codes2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 19, p. 2973-2977Article in journal (Refereed)
    Abstract [en]

    It is proved that the size of the symmetry group Sym(C) of every full rank perfect 1-error correcting binary code C of length n is less than or equal to 2|Sym( Hn)|(n+1), where Hn is a Hamming code of the same length.

  • 10.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A remark on full rank perfect codes2006In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 306, no 16, p. 1975-1980Article in journal (Refereed)
    Abstract [en]

    Any full rank perfect 1-error correcting binary code of length n = 2(k) - 1 and with a kernel of dimension n - log(n + 1) - m, where in is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2(m) - 1 and with a kernel of dimension n - log(n + 1) - k. Especially we may construct full rank perfect 1-error correcting binary codes of length n = 2(m) - 1 and with a kernel of dimension n - log(n + 1) - 4 for nt = 6, 7,..., 10. This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.

  • 11.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A survey of perfect codes2008Article, review/survey (Refereed)
    Abstract [en]

    The first examples of perfect e-error correcting q-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietavainen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect q-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem. This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

  • 12.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Full rank perfect codes and alpha-kernels2009In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 8, p. 2202-2216Article in journal (Refereed)
    Abstract [en]

    A perfect 1-error correcting binary code C, perfect code for short, of length n = 2(m) - 1 has full rank if the linear span < C > of the words of C has dimension n as a vector space over the finite field F-2. There are just a few general constructions of full rank perfect codes, that are not given by recursive methods using perfect codes of length shorter than n. In this study we construct full rank perfect codes, the so-called normal alpha-codes, by first finding the superdual of the perfect code. The superdual of a perfect code consists of two matrices G and T in which simplex codes play an important role as subspaces of the row spaces of the matrices G and T. The main idea in our construction is the use of alpha-words. These words have the property that they can be added to certain rows of generator matrices of simplex codes such that the result will be (other) sets of generator matrices for simplex codes. The kernel of these normal alpha-codes will also be considered. It will be proved that any subspace, of the kernel of a normal alpha-code, that satisfies a certain property will be the kernel of another perfect code, of the same length n. In this way, we will be able to relate some of the full rank perfect codes of length n to other full rank perfect codes of the same length n.

  • 13.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Linear maps of perfect codes and irregular C-partitions2015In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 338, no 3, p. 149-163Article in journal (Refereed)
    Abstract [en]

    The concept of an irregular C-partition of binary space into perfect 1-error-correcting codes is defined. Three distinct constructions of irregular C-partitions are presented. The relation between irregular C-partitions and linear maps, that map perfect codes to perfect codes, is discussed

  • 14.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    Maximal partial spreads and the modular n-queen problem III2002In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 243, no 3-Jan, p. 135-150Article in journal (Refereed)
    Abstract [en]

    Maximal partial spreads in PG(3, q) q = p(k), p odd prime and q greater than or equal to 7, are constructed for any integer n in the interval (q(2) + 1)/2 + 6 less than or equal to n less than or equal to (5q(2) + 4q - 1)/8 in the case q + 1 0, +/-2, +/-4, +/-6, +/-10, 12 (mod 24). In all these cases. maximal partial spreads of the size (q(2) + 2 + n have also been constructed for some small values of the integer n. These values depend on q and are mainly n = 3 and n = 4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3, q), q = p(k) where p is an odd prime and q greater than or equal to 7, of size n for any integer n in the interval (q(2) + 1) /2 + 6 less than or equal to n less than or equal to q(2) - q + 2.

  • 15.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    Maximal partial spreads in PG(3,5)2000In: Ars combinatoria, ISSN 0381-7032, Vol. 57, p. 97-101Article in journal (Refereed)
    Abstract [en]

    Maximal partial spreads of the sizes 13, 14, 15,..., 22 and 26 are described. They were found by using a computer. The computer also made a complete search for maximal partial spreads of size less then or equal to 12. No such maximal partial spreads were found.

  • 16.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Necessary and sufficient conditions for the existence of a class of partitions of a finite vector space2009In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 53, no 2, p. 69-73Article in journal (Refereed)
    Abstract [en]

    Necessary and sufficient conditions for the existence of a partition of a finite vector space over the finite field GF(p), where p is a prime, into subspaces where all but p of the subspaces have the same dimension, are presented.

  • 17.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    No maximal partial spread of size 115 in PG(3,11)2003In: Ars combinatoria, ISSN 0381-7032, Vol. 66, p. 139-155Article in journal (Refereed)
    Abstract [en]

    It is proved that there is no maximal partial spread of size 115 in PG(3,11).

  • 18.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On kernels of perfect codes2010In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 310, no 21, p. 3052-3055Article in journal (Refereed)
    Abstract [en]

    It is shown that there exists a perfect one-error-correcting binary code with a kernel which is not contained in any Hamming code.

  • 19.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On perfect codes over non prime power alphabets2010In: ERROR-CORRECTING CODES, FINITE GEOMETRIES AND CRYPTOGRAPHY / [ed] Bruen AA; Wehlau DL, 2010, Vol. 523, p. 173-184Conference paper (Refereed)
    Abstract [en]

    Known results on perfect codes over alphabets with q elements where q is not a prime power is surveyed. Some tools recently developed for the study of perfect 1-error correcting binary codes are generalized to the case of non prime power alphabets. A theorem by H. W. Lenstra from 1972 on the algebraic structure of perfect codes over non prime power alphabets will, by using this generalization, be further strengthened.

  • 20.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On perfect p-ary codes of length p+12008In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 46, no 1, p. 45-56Article in journal (Refereed)
    Abstract [en]

    Let p be a prime number and assume p >= 5. We will use a result of L. Redei to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p - 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.

  • 21.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the length of the tail of a vector space partition2009In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 21, p. 6169-6180Article in journal (Refereed)
    Abstract [en]

    A vector space partition P of a finite dimensional vector space V = V(n, q) of dimension n over a finite field with q elements, is a collection of subspaces U-1, U-2, ..., U-t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of P consists of the subspaces of least dimension d(1) in P, and the length n(1) of the tail is the number of subspaces in the tail. Let d(2) denote the second least dimension in P. Two cases are considered: the integer q(d2-d1) does not divide respective divides n(1). In the first case it is proved that if 2d(1) > d(2) then n(1) >= q(d1) + 1 and if 2d(1) <= d(2) then either n(1) = (q(d2) - 1)/(q(d1) - 1) or n(1) > 2q(d2-d1). These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2d(1) respectively d(2). In case q(d2-d1) divides n(1) it is shown that if d(2) < 2d(1) then n(1) >= q(d2) - q(d1) + q(d2-d1) and if 2d(1) <= d(2) then n(1) <= qd(2.) The last bound is also shown to be tight. The results considerably improve earlier found lower bounds on the length of the tail.

  • 22.
    Heden, Olof
    KTH, Superseded Departments, Mathematics.
    On the reconstruction of perfect codes2002In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 256, no 2-Jan, p. 479-485Article in journal (Refereed)
    Abstract [en]

    We show how to reconstruct a perfect I-error correcting binary code of length n from the code words of weight (n + 1)/2.

  • 23.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the size of the symmetry group of a perfect code2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 17, p. 1879-1885Article in journal (Refereed)
    Abstract [en]

    It is shown that for every nonlinear perfect code C of length n and rank r with n - log(n + 1) + 1 <= r <= n - 1, vertical bar Sym(C)vertical bar <= vertical bar GL(n - r, 2)vertical bar . vertical bar GL(log(n +1) - (n - r), 2)vertical bar . (n + 1/2(n-r))(n-r) where Sym(C) denotes the group of symmetries of C. This bound considerably improves a bound of Malyugin. (C) 2011 Elsevier B.V. All rights reserved.

  • 24.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Perfect codes from the dual point of view I2008In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 308, no 24, p. 6141-6156Article in journal (Refereed)
    Abstract [en]

    The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.

  • 25.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    PERFECT CODES OF LENGTH n WITH KERNELS OF DIMENSION n - log(n+1)-32008In: SIAM Journal on Discrete Mathematics, ISSN 0895-4801, E-ISSN 1095-7146, Vol. 22, no 4, p. 1338-1350Article in journal (Refereed)
    Abstract [en]

    Perfect 1-error correcting binary codes are considered. Those of length n and with a kernel of dimension n - log(n + 1) - 3 are shown to be obtainable by the Phelps construction.

  • 26.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The partial order of perfect codes associated to a perfect code2007In: Advances in Mathematics of Communications, ISSN 1930-5346, Vol. 1, no 4, p. 399-412Article in journal (Refereed)
    Abstract [en]

    It is clarified whether or not "full rank perfect 1-error correcting binary codes act like primes in the family of all perfect 1-error correcting binary codes". Thereby the well known connection between perfect 1-error correcting binary codes and tilings will be discussed and used.

  • 27.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Guzeltepe, Murat
    On perfect 1-epsilon-error-correcting codes2015In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 20, no 1, p. 23-35Article in journal (Refereed)
    Abstract [en]

    We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-epsilon-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting q-ary Hamming codes as subsets.

  • 28.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Guzeltepe, Murat
    Perfect 1-error-correcting Lipschitz weight codes2016In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 21, no 1, p. 23-30Article in journal (Refereed)
    Abstract [en]

    Let pi be a Lipschitz prime and p = pi pi(star). Perfect 1-error-correcting codes in H(Z)(n)(pi), are constructed for every prime number p equivalent to 1(mod 4). This completes a result of the authors in an earlier work, Perfect Mannheim, Lipschitz and Hurwitz weight codes, (Math. Commun. 19(2014), 253-276), where a construction is given in the case p 3 (mod 4).

  • 29.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Güzeltepe, M.
    On perfect 1-ε-error-correcting codes2015In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 20, no 1, p. 23-35Article in journal (Refereed)
    Abstract [en]

    We generalize the concept of perfect Lee-error-correcting codes, and present constructions of this new class of perfect codes that are called perfect 1-ε-error-correcting codes. We also show that in some cases such codes contain quite a few perfect 1-error-correcting q-ary Hamming codes as subsets.

  • 30.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hessler, Martin
    On linear equivalence and phelps codes2010In: Advances in Mathematics of Communications, ISSN 1930-5346, Vol. 4, no 1, p. 69-81Article in journal (Refereed)
    Abstract [en]

    It is shown that all non-full-rank FRH-codes, a class of perfect codes we define in this paper, are linearly equivalent to perfect codes obtainable by Phelps' construction. Moreover, it is shown by an example that the class of perfect FRH-codes also contains perfect codes that are not obtainable by Phelps construction.

  • 31.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hessler, Martin
    On linear equivalence and phelps codes. Addendum2011In: Advances in Mathematics of Communications, ISSN 1930-5346, Vol. 5, no 3, p. 543-546Article in journal (Refereed)
    Abstract [en]

    A new class of perfect 1-error correcting binary codes, so called RRH-codes, are identified, and it is shown that every such code is linearly equivalent to a perfect code obtainable by the Phelps construction.

  • 32.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hessler, Martin
    On the classification of perfect codes: side class structures2006In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 40, no 3, p. 319-333Article in journal (Refereed)
    Abstract [en]

    The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C' are linearly equivalent if there exists a non-singular matrix A such that AC = C' where C and C' are matrices with the code words of C and C' as columns. Hessler proved that the perfect codes C and C' are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.

  • 33.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hessler, Martin
    Westerbäck, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the classification of perfect codes: Extended side class structures2010In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 310, no 1, p. 43-55Article in journal (Refereed)
    Abstract [en]

    The two 1-error correcting perfect binary codes. C and C' are said to be equivalent if there exists a permutation pi of the set of the n coordinate positions and a word (d) over bar such that C' = pi((d) over bar + C). Hessler defined C and C' to be linearly equivalent if there exists a non-singular linear map phi such that C' = phi(C). Two perfect codes C and C' of length n will be defined to be extended equivalent if there exists a non-singular linear map W and a word (d) over bar such that C' = phi((d) over bar + C). Heden and Hessler, associated with each linear equivalence class an invariant L-C and this invariant was shown to be a subspace of the kernel of some perfect code. It is shown here that, in the case of extended equivalence, the corresponding invariant will be the extension of the code L-C. This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.

  • 34.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Krotov, Denis S.
    ON THE STRUCTURE OF NON-FULL-RANK PERFECT q-ARY CODES2011In: ADVANCES IN MATHEMATICS OF COMMUNICATIONS, ISSN 1930-5346, Vol. 5, no 2, p. 149-156Article in journal (Refereed)
    Abstract [en]

    The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-error-correcting binary code can be constructed by this combining construction is generalized to the q-ary case. Simply speaking, every non-full-rank perfect code C is the union of a well-defined family of (mu) over bar -components K((mu) over bar), where (mu) over bar belongs to an "outer" perfect code C(star), and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain (mu) over bar -components, and new lower bounds on the number of perfect 1-error-correcting q-ary codes are presented.

  • 35.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lehmann, J.
    Nastase, E.
    Sissokho, P.
    On the type(s) of minimum size subspace partitions2014In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 332, p. 1-9Article in journal (Refereed)
    Abstract [en]

    Let V = V(kt + r, q) be a vector space of dimension kt + r over the finite field with q elements. Let sigma(q)(kt + r, t) denote the minimum size of a subspace partition P of V in which t is the largest dimension of a subspace. We denote by n(di) the number of subspaces of dimension d(i) that occur in P and we say [d(1)(nd1),..., d(m)(ndm)] is the type of P. In this paper, we show that a partition of minimum size has a unique partition type if t + r is an even integer. We also consider the case when t + r is an odd integer, but only give partial results since this case is indeed more intricate.

  • 36.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lehmann, J.
    Nastase, E.
    Sissokho, P.
    The supertail of a subspace partition2013In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 69, no 3, p. 305-316Article in journal (Refereed)
    Abstract [en]

    Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d (1) < d (2) < ... < d (m) be the dimensions that occur in a subspace partition of V. Let sigma (q) (n, t) denote the minimum size of a subspace partition of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s a parts per thousand currency sign m, the set of subspaces in of dimension less than d (s) is called the s-supertail of . The main result is that the number of spaces in an s-supertail is at least sigma (q) (d (s) , d (s-1)).

  • 37.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lehmann, Juliane
    Univ Bremen, Bremen, Germany .
    Nastase, Esmeralda
    Xavier Univ, Cincinnati, OH USA .
    Sissokho, Papa
    Illinois State Univ, Normal, IL USA .
    Extremal sizes of subspace partitions2012In: Designs, Codes and Cryptography, ISSN 0925-1022, E-ISSN 1573-7586, Vol. 64, no 3, p. 265-274Article in journal (Refereed)
    Abstract [en]

    A subspace partition I of V = V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of I . The size of I is the number of its subspaces. Let sigma (q) (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let rho (q) (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of sigma (q) (n, t) and rho (q) (n, t) for all positive integers n and t. Furthermore, we prove that if n a parts per thousand yen 2t, then the minimum size of a maximal partial t-spread in V(n + t -1, q) is sigma (q) (n, t).

  • 38.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Marcugini, S.
    Pambianco, F.
    Storme, L.
    On the non-existence of a maximal partial spread of size 76 in PG(3,9)2008In: Ars combinatoria, ISSN 0381-7032, Vol. 89, p. 369-382Article in journal (Refereed)
    Abstract [en]

    We prove the non-existence of maximal partial spreads of size 76 in PG(3,9). Relying on the classification of the minimal blocking sets of size 15 in PG(2, 9) [22], we show that there are only two possibilities for the set of holes of such a maximal partial spread. The weight argument of Blokhuis and Metsch [3] then shows that these sets cannot be the set of holes of a maximal partial spread of size 76. In [17], the non-existence of maximal partial spreads of size 75 in PG(3,9) is proven. This altogether proves that the largest maximal partial spreads, different from a spread, in PG(3, q = 9) have size q(2) - q + 2 = 74.

  • 39.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pasticci, Fabio
    Westerbäck, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP2009In: Advances in Mathematics of Communications, ISSN 1930-5346, Vol. 3, no 3, p. 295-309Article in journal (Refereed)
    Abstract [en]

    The set of permutations of the coordinate set that maps a perfect code C into itself is called the symmetry group of C and is denoted by Sym(C). It is proved that for all integers n = 2(m) - 1, where m = 4, 5, 6, ... , and for any integer r, where n - log(n + 1) + 3 <= r <= n - 1, there are perfect codes of length n and rank r with a trivial symmetry group, i.e. Sym(C) = {id}. The result is shown to be true, more generally, for the extended perfect codes of length n + 1.

  • 40.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pasticci, Fabio
    Univ Perugia.
    Westerbäck, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    on the symmetry group of extended perfect binary codes of length n+1 and rank n-log(n+1)+22012In: Advances in Mathematics of Communication, ISSN 1930-5346, Vol. 6, no 2, p. 121-130Article in journal (Refereed)
    Abstract [en]

    It is proved that for every integer n = 2(k) - 1, with k >= 5, there exists a perfect code C of length n, of rank r = n - log(n + 1) + 2 and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length n = 2(k) - 1, with k >= 5, and any rank r, with n - log(n + 1) + 3 <= r <= n - 1 there exist perfect codes with a trivial symmetry group.

  • 41.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Roos, Cornelis
    The non-existence of some perfect codes over non-prime power alphabets2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 14, p. 1344-1348Article in journal (Refereed)
    Abstract [en]

    Let exp(p)(q) denote the number of times the prime number p appears in the prime factorization of the integer q. The following result is proved: If there is a perfect 1-error correcting code of length n over an alphabet with q symbols then, for every prime number p. exp(p)(1 + n(q - 1)) <= exp(p)(q)(1 + (n - 1)/q). This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters (n q, e) = (19, 6. 1).

  • 42.
    Heden, Olof
    et al.
    KTH, Superseded Departments, Mathematics.
    Saggese, M.
    Bruen chains in PG(3, p(k)) k >= 22000In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 214, no 3-Jan, p. 251-253Article in journal (Refereed)
    Abstract [en]

    The Bruen chains of PC(3,q) for q = 9, 25 and 27 are described. They were found by a computer search. In the case q = 49 no chains have been found yet.

  • 43.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sissokho, Papa A.
    On the existence of a (2,3)-spread in V(7,2)2016In: Ars combinatoria, ISSN 0381-7032, Vol. 124, p. 161-164Article in journal (Refereed)
    Abstract [en]

    An (s, t)-spread in a finite vector space V = V (n, q) is a collection F of t-dimensional subspaces of V with the property that every s-dimensional subspace of V is contained in exactly one member of F. It is remarkable that no (s, t)-spreads has been found yet, except in the case s = 1. In this note, the concept a-point to a (2,3)-spread F in V = V(7, 2) is introduced. A classical result of Thomas, applied to the vector space V, states that all points of V cannot be alpha-points to a given (2, 3)-spread.F. in V. In this note, we strengthened this result by proving that every 6-dimensional subspace of V must contain at least one point that is not an a-point to a given (2, 3)-spread of V.

  • 44.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solov'eva, Faina I.
    PARTITIONS OF F-n INTO NON-PARALLEL HAMMING CODES2009Article in journal (Refereed)
    Abstract [en]

    We investigate partitions of the set F-n of all binary vectors of length n into cosets of pairwise distinct linear Hamming codes ("non-parallel Hamming codes") of length n. We present several constructions of partitions of F-n into non-parallel Hamming codes of length n and discuss a lower bound on the number of different such partitions.

  • 45.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solov'evay, F. I.
    Mogilnykhy, I.Yu.
    Intersections of perfect binary codes2010In: Proceedings - 2010 IEEE Region 8 International Conference on Computational Technologies in Electrical and Electronics Engineering, SIBIRCON-2010, 2010, p. 52-54Conference paper (Refereed)
    Abstract [en]

    Intersections of perfect binary codes are investigated. In 1998 Etzion and Vardy proved that the intersection number η(C;D), for any two distinct perfect codes C and D, is always in the range 0 ≤η(C;D) ≤2 n-log(n+1)-2(n-1)/2; where the upper bound is attainable. We improve the upper bound and show that the intersection number 2n-log(n+1) -2(n-1)/2 is "sporadic". We also find a large class of intersection numbers for perfect binary codes of length 15 and for any admissible n &gt;15 a new set of intersection numbers for perfect codes of length n.

  • 46.
    Heden, Olof
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Westerbäck, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non Phelps codes of length 15 and rank 142007In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 38, p. 141-153Article in journal (Refereed)
  • 47. Lehmann, Juliane
    et al.
    Heden, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Some necessary conditions for vector space partitions2012In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 312, no 2, p. 351-361Article in journal (Refereed)
    Abstract [en]

    Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V(2t, q) that contains m(d) spaces of dimension d, where t/2 < d < t, and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t-spreads in V (2t, q). We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.

1 - 47 of 47
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