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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: 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: 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: 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: 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]

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]

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]