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A mapping of k -bit strings into n -bit strings is called an ( α , β ) -map if k -bit strings which are more than αk apart are mapped to n -bit strings that are more than βn apart in Hamming distance. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to α = 0 . Existence of an ( α , β ) -map is equivalent to existence of a graph homomorphism H ¯ ( k , α k ) → H ¯ ( n , β n ) , where H ( n , d ) is a Hamming graph with vertex set { 0 , 1 } n and edges connecting vertices differing in d or fewer entries. This paper proves impossibility results on achievable parameters ( α , β ) in the regime of n , k → ∞ with a fixed ratio n k = ρ . This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's θ -function). The criterion is then evaluated using some known and some new results from coding theory concerning the θ -function of Hamming graphs. As an example, it is shown that if β > 1 / 2 and n k – integer, the n k -fold repetition map achieving α = β is asymptotically optimal. Finally, constraints on configurations of points and hyperplanes in projective spaces over F 2 are derived. [ABSTRACT FROM AUTHOR]

This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q -analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric. [ABSTRACT FROM AUTHOR]

Abstract: Traceability codes are combinatorial objects introduced by Chor, Fiat and Naor in 1994 to be used in traitor tracing schemes to protect digital content. A k-traceability code is used in a scheme to trace the origin of digital content under the assumption that no more than k users collude. It is well known that an error correcting code of high minimum distance is a traceability code. When does this ‘error correcting construction’ produce good traceability codes? The paper explores this question. Let ℓ be a fixed positive integer. When q is a sufficiently large prime power, a suitable Reed–Solomon code may be used to construct a 2-traceability code containing codewords. The paper shows that this construction is close to best possible: there exists a constant c, depending only on ℓ, such that a q-ary 2-traceability code of length ℓ contains at most codewords. This answers a question of Kabatiansky from 2005. Barg and Kabatiansky (2004) asked whether there exist families of k-traceability codes of rate bounded away from zero when q and k are constants such that . These parameters are of interest since the error correcting construction cannot be used to construct k-traceability codes of constant rate for these parameters: suitable error correcting codes do not exist when because of the Plotkin bound. Kabatiansky (2004) answered Barg and Kabatiansky''s question (positively) in the case when . This result is generalised to the following: whenever k and q are fixed integers such that and , or such that and , there exist infinite families of q-ary k-traceability codes of constant rate. [Copyright &y& Elsevier]

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the F q m -linear MRD codes, the F q -linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of F q -linear MRD codes in F q n × m in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q → + ∞ and as m → + ∞. We also prove that MDS and F q m -linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of F q -linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code. [ABSTRACT FROM AUTHOR]