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  • 1. Adamaszek, Michal
    et al.
    Barmak, Jonathan Ariel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On a lower bound for the connectivity of the independence complex of a graph2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 21, p. 2566-2569Article in journal (Refereed)
    Abstract [en]

    Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.

  • 2.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Panova, Greta
    LLT polynomials, chromatic quasisymmetric functions and graphs with cycles2018In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 341, no 12, p. 3453-3482Article in journal (Refereed)
    Abstract [en]

    We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the noncircular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall-Littlewood polynomials. 

  • 3.
    Amini, Nima
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Alexandersson, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The cone of cyclic sieving phenomena2019In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, no 6, p. 1581-1601Article in journal (Refereed)
    Abstract [en]

    We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

    Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

  • 4. Bränden, Petter
    q-Narayana numbers and the flag h-vector2004In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 281, no 03-jan, p. 67-81Article in journal (Refereed)
    Abstract [en]

    The Narayana numbers are N(n,k) = (1/n)((n)(k))((n)(k+1)). There are several natural statistics on Dyck paths with a distribution given by N(n, k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J(2 x n) in different ways. In the process we discover new Narayana statistics and provide co-statistics for the Narayana statistics so that the bi-statistics have a distribution given by Furlinger and Hofbauer's q-Narayana numbers. We interpret the flag h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions. We also introduce what we call pre-shellings of simplicial complexes.

  • 5. Bränden, Petter
    et al.
    Claesson, A.
    Steingrimsson, E.
    Catalan continued fractions, and increasing subsequences in permutations2002In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 258, no 03-jan, p. 275-287Article in journal (Refereed)
    Abstract [en]

    We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e(k)(pi) be the number of increasing subsequences of length k + 1 in the permutation pi. We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the e(k)S. Moreover, there is an invertible linear transformation that translates between linear combinations of ekS and the corresponding continued fractions. Some applications are given, one of which relates fountains of coins to 132-avoiding permutations according to number of inversions. Another relates ballot numbers to such permutations according to number of right-to-left maxima.

  • 6.
    Engström, Alexander
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Complexes of directed trees and independence complexes2009In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 10, p. 3299-3309Article in journal (Refereed)
    Abstract [en]

    First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied to generalize the results by Kozlov.An interesting special case is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of R the homotopy type of the anti-Rips complex is determined.

  • 7.
    Eriksson, Henrik
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Eriksson, Kimmo
    Karlander, Johan
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Svensson, Lars Erik
    KTH, Superseded Departments, Mathematics.
    Wästlund, Johan
    KTH, Superseded Departments, Mathematics.
    Sorting a bridge hand2001In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 241, no 1-3, p. 289-300Article in journal (Refereed)
    Abstract [en]

    Sorting a permutation by block moves is a task that every bridge player has to solve every time she picks up a new hand of cards. It is also a problem for the computational biologist, for block moves are a fundamental type of mutation that can explain why genes common to two species do not occur in the same order in the chromosome, It is not known whether there exists an optimal sorting procedure running in polynomial time. Bafna and Pevzner gave a polynomial time algorithm that sorts any permutation of length n in at most 3n/4 moves. Our new algorithm improves this to [(2n - 2)/3] for n greater than or equal to 9. For the reverse permutation, we give an exact expression for the number of moves needed, namely [(n + 1)/2]. Computations of Bafha and Pevzner up to n = 10 seemed to suggest that this is the worst case; but as it turns out, a first counterexample occurs for n = 13, i.e. the bridge player's case. Professional card players never sort by rank, only by suit. For this case, we give a complete answer to the optimal sorting problem.

  • 8. Eriksson, K.
    et al.
    Karlander, Johan
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Stable matching in a common generalization of the marriage and assignment models2000In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 217, no 3-Jan, p. 135-156Article in journal (Refereed)
    Abstract [en]

    In the theory of two-sided matching markets there are two well-known models: the marriage model (where no money is involved) and the assignment model (where payments are involved). Roth and Sotomayor, Two-Sided Matching, Cambridge University Press, Cambridge, 1990, asked for an explanation for the similarities in behavior between those two models. We address this question by introducing a common generalization that preserves the two important features: the existence of a stable outcome and the lattice property of the set of stable outcomes.

  • 9. Guibert, O.
    et al.
    Linusson, Svante
    Doubly alternating Baxter permutations are Catalan2000In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 217, no 03-jan, p. 157-166Article in journal (Refereed)
    Abstract [en]

    The Baxter permutations who are alternating and whose inverse is also alternating are shown to be enumerated by the Catalan numbers. A bijection to complete binary trees is also given.

  • 10.
    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.

  • 11.
    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.

  • 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, 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.

  • 16.
    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.

  • 17.
    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.

  • 18.
    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.

  • 19.
    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.

  • 20.
    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.

  • 21.
    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.

  • 22.
    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).

  • 23.
    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.

  • 24.
    Hessler, Martin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Perfect codes as isomorphic spaces2006In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 306, no 16, p. 1981-1987Article in journal (Refereed)
    Abstract [en]

    Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.

  • 25.
    Hultman, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The topology of spaces of phylogenetic trees with symmetry2007In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 307, no 14, p. 1825-1832Article in journal (Refereed)
    Abstract [en]

    Natural Dowling analogues of the complex of phylogenetic trees are studied. Using discrete Morse theory, we find their homotopy types. In the process, the homotopy types of certain subposets of Dowling lattices are determined.

  • 26. 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.

  • 27. Linusson, Svante
    Extended pattern avoidance2002In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 246, no 03-jan, p. 219-230Article in journal (Refereed)
    Abstract [en]

    A 0-1 matrix is said to be extendably tau-avoiding if it can be the upper left corner of a tau-avoiding permutation matrix. This concept arose in Eriksson and Linusson (Electron. J. Combin. 2 (1995) R6) where the surprising result that the number of extendably 321-avoiding rectangles are enumerated by the ballot numbers was proved, Here we study the other five patterns of length three. The main result is that the six patterns of length three divide into only two cases, no easy symmetry can explain this. Another result is that the Simion-Schmidt-West bijection for permutations avoiding patterns 12tau and 21tau works also for extended pattern avoidance. As an application, we use the results on extended pattern avoidance to prove a sequence of refinements on the enumeration of permutations avoiding patterns of length 3. The results and proofs use many properties and refinements of the Catalan numbers.

  • 28.
    Lundow, Per Håkan
    Umeå University, Department of Mathematics.
    Compression of transfer matrices2001In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 231, no 03-jan, p. 321-329Article in journal (Refereed)
    Abstract [en]

    We present a method for reducing the size of transfer matrices by exploiting symmetry. For example, the transfer matrix for enumeration of matchings in the graph C-4 x C-4 x P-n can be reduced from order 65536 to 402 simply due to the 384 automorphisms of C-4 x C-4. The matrix for enumeration of perfect matchings can be still further reduced to order 93, all in a straightforward and mechanical way. As an application we report an improved upper bound for the three-dimensional dimer problem.

  • 29. Pasticci, Fabio
    et al.
    Westerbäck, Thomas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On rank and kernel of some mixed perfect codes2009In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 309, no 9, p. 2763-2774Article in journal (Refereed)
    Abstract [en]

    Mixed perfect 1-error correcting codes and the associated dual codes over the group Z (n, l), Z (n, l) = under(under(Z2 × Z2 × ⋯ × Z2, {presentation form for vertical right curly bracket}), n) × underover(Z, 2, l), n ≥ 1 and l ≥ 2, are investigated. A lower and an upper bound for the rank k of the kernel of mixed perfect 1-error correcting codes in Z (n, l), depending on the rank r of the mixed perfect code and the structure of the corresponding dual code, are given. Due to a general construction of mixed perfect 1-error correcting group codes in Z (n, l), we show that the upper bound is tight for some n, l and r.

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