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In this paper, we introduce some new polynomials associated to linear codes over F q. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over F q , and give the MacWilliams type identity for it. We also give the notion of generalized q -colored t -designs. As an application of the generalized q -colored t -designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over F q. Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type III and Type IV codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Jacobi polynomials of a code with multiple reference vectors, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs. [ABSTRACT FROM AUTHOR]

Abstract: Linear codes arising from the row span over any prime field of the incidence matrices of the odd graphs for are examined and all the main parameters obtained. A study of the hulls of these codes for yielded that for (the Petersen graph), the dual of the binary hull from an incidence matrix is the binary code from points and lines of the projective geometry , which leads to a correspondence between the edges and vertices of with the points and a collection of ten lines of , consistent with the codes. The study also gives the dimension, the minimum weight, and the nature of the minimum words, of the binary codes from adjacency matrices of the line graphs . [Copyright &y& Elsevier]

Abstract: The enumeration of strongly regular graphs with parameters (45, 12, 3, 3) has been completed, and it is known that there are 78 non-isomorphic strongly regular (45, 12, 3, 3) graphs. A strongly regular graph with these parameters is a symmetric (45, 12, 3) design having a polarity with no absolute points. In this paper we examine the ternary codes obtained from the adjacency (resp. incidence) matrices of these graphs (resp. designs), and those of their corresponding derived and residual designs. Further, we give a generalization of a result of Harada and Tonchev on the construction of non-binary self-orthogonal codes from orbit matrices of block designs under an action of a fixed-point-free automorphism of prime order. Using the generalized result we present a complete classification of self-orthogonal ternary codes of lengths 12, 13, 14, and 15, obtained from non-fixed parts of orbit matrices of symmetric (45, 12, 3) designs admitting an automorphism of order 3. Several of the codes obtained are optimal or near optimal for the given length and dimension. We show in addition that the dual codes of the strongly regular (45, 12, 3, 3) graphs admit majority logic decoding. [Copyright &y& Elsevier]

Abstract: We study the distribution of triples of codewords of codes and ordered codes. Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (8) (2005) 2859–2866] used the triple distribution of a code to establish a bound on the number of codewords based on semidefinite programming. In the first part of this work, we generalize this approach for ordered codes. In the second part, we consider linear codes and linear ordered codes and present a MacWilliams-type identity for the triple distribution of their dual code. Based on the non-negativity of this linear transform, we establish a linear programming bound and conclude with a table of parameters for which this bound yields better results than the standard linear programming bound. [Copyright &y& Elsevier]

Abstract: We examine the -ary linear codes from incidence matrices of the three uniform subset graphs with vertex set the set of subsets of size 3 of a set of size , with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. All the main parameters of the codes and the nature of the minimum words are obtained, and it is shown that the codes can be used for full error-correction by permutation decoding. We examine also the binary codes of the line graphs of these graphs. [Copyright &y& Elsevier]

Abstract: We show that the first- and second-order Reed–Muller codes, and , can be used for permutation decoding by finding, within the translation group, - and -PD-sets for for , respectively, and -PD-sets for for . We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967–1970]. [Copyright &y& Elsevier]

Abstract: We examine the -ary codes, for any prime , that can be obtained from incidence matrices and line graphs of the Hamming graphs, , obtaining the main parameters of these codes. We show that the codes from the incidence matrices of can be used for full permutation decoding for all . [Copyright &y& Elsevier]

Abstract: Almost difference families (ADFs) are a useful generalization of almost difference sets (ADSs). In this paper, we present some constructive techniques to obtain ADFs and establish a number of infinite classes of ADFs. Our results can be regarded as a generalization of the known difference families. It is clear that ADFs give partially balance incomplete block designs which arise in a natural way in many combinatorial and statistical problems. [Copyright &y& Elsevier]

Abstract: We prove for a large class of parameters t and r that a multiset of at most points in that blocks every k-dimensional subspace at least t times must contain a sum of t subspaces of codimension k. We use our results to identify a class of parameters for linear codes for which the Griesmer bound is not sharp. Our theorem generalizes the non-existence results from Maruta [On the achievement of the Griesmer bound, Des. Codes Cryptogr. 12 (1997) 83–87] and Klein [On codes meeting the Griesmer bound, Discrete Math. 274 (2004) 289–297]. [Copyright &y& Elsevier]

Abstract: Rosenbloom and Tsfasman introduced a generalization of the Hamming metric, motivated by a model of information transmission over parallel channels, and obtained bounds on the cardinality of codes in the induced finite metric space. In the present paper, some new bounds are given and the analogue of the Plotkin bound is improved. Furthermore, the following problem together with some results is presented: which are the parameters such that there exist codes meeting the Singleton bound in the new space? The binary case is solved. [Copyright &y& Elsevier]

Abstract: The binary codes of the line graphs of the complete multipartite graphs ( for ) , are examined, and PD-sets and s-PD-sets are found. [Copyright &y& Elsevier]

Abstract: An integer-valued function on the integers that is periodic of period , p prime, can be matched, modulo , by a polynomial function ; we show that may be taken to have degree at most . Applications include a short proof of the theorem of McEliece on the divisibility of weights of codewords in p-ary cyclic codes by powers of p, an elementary proof of the Ax–Katz theorem on solutions of congruences modulo p, and results on the numbers of codewords in p-ary linear codes with weights in a given congruence class modulo . [Copyright &y& Elsevier]

Abstract: Every elliptic quartic of with n -rational points provides a near-MDS code of length n and dimension 4 such that the collineation group of is isomorphic to the automorphism group of . In this paper we assume that has characteristic . We classify the linear collineation groups of which can preserve an elliptic quartic of . Also, we prove for that if the j-invariant of does not disappear, then cannot be extended in a natural way by adding a point of to . [Copyright &y& Elsevier]

For a set Ω of size n⩾7 and Ω{3} the set of subsets of Ω of size 3, we examine the binary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω{3}, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. [Copyright &y& Elsevier]

The notion of t-design in a Grassmannian space Gm,n was introduced by the first and last authors and G. Nebe in a previous paper. In the present work, we give a general lower bound for the size of such designs. The method is inspired by Delsarte, Goethals and Seidel work in the case of spherical designs. This leads us to introduce a notion of f-code in Grassmannian spaces, for which we obtain upper bounds, as well as a kind of duality tight-designs/tight-codes. The bounds are in terms of the dimensions of the irreducible representations of the orthogonal group O(n) occurring in the decomposition of the space L2(Gm,n°) of square integrable functions on Gm,n°, the set of oriented Grassmanianns. [Copyright &y& Elsevier]

We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of error-correcting codes. Finally, we give necessary and sufficient conditions for the injective chromatic number to be equal to the degree for a regular graph. [Copyright &y& Elsevier]

For a finite group G , a positive integer λ , and subsets X , Y of G , write λ G = X Y if the products x y (x ∈ X , y ∈ Y), cover G precisely λ times. Such a subset Y is called a code with respect to X , and when λ = 1 it is a perfect code in the Cayley graph Cay (G , X). In this paper we present various families of examples of such codes, with X closed under conjugation and Y a subgroup, in symmetric groups, and also in special linear groups S L 2 (q). We also propose conjectures about the existence of some much wider families. [ABSTRACT FROM AUTHOR]

We show there is a bijection between the binary necklaces with n black beads and k white beads and certain (n , k) -codes when n is prime. The main idea is to come up with a new map on necklaces called slime migration. [ABSTRACT FROM AUTHOR]