We consider the problem of defining and evaluating the Hamming distance between two– dimensional pattern P [1..m, 1..m] of pixels and two–dimensional text T [1..n, 1..n] of pixels when also rotations of P are allowed. In particular, we are interested in the orientation and location of P that gives the minimum Hamming distance. The number of different orientations for P is O(m3) when the center ...

We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised Cab polynomials the new bound often improves dramatically on the Feng-Rao bound for primary codes [1, 10]. The method does not only work for the minimum distance but can be applied to any generalised Hamming weight.

In this note, we investigate the Hamming weight enumerators of self-dual codes over F q and Zk. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong for self-dual codes over F q and Zk is determined. 2

If Fq is a finite field and G is a subgroup of the linear automorphisms of F q , a solution to the problem of finding all the G-invariant linear codes C of F q (i.e. such that g(C) = C for all g ∈ G) is offered. This will be referred as the invariance problem. When n = |G|t, we determine conditions for the existance of an isomorphism of Fq[G]modules between F q and Fq[G]× · · · × Fq[G] (t-times...

The Wiener number W (G) of a graph G is the sum of distances between all pairs of vertices of G. If (G,w) is a vertex-weighted graph, then the Wiener number W (G,w) of (G,w) is the sum, over all pairs of vertices, of products of weights of the vertices and their distance. For G being a partial binary Hamming graph, a formula is given for computing W (G,w) in terms of a binary Hamming labeling o...

It is established that for any q > 2 the permutation automorphism group of a q-ary Hamming code of length n = (q − 1)/(q− 1) is isomorphic to the unitriangular group UTm(q).

In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group Sn and are consistent as n varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity. MSC:

We present an extension of the pairwise Hamming distance, the r-wise Hamming distance, and show that it can be used to fully characterize the maximum-likelihood decoding (MLD) error of an arbitrary code used over the binary erasure channel (BEC). Based on these insights, we present a new design criterion for a code: the minimum r-wise Hamming distance. We prove that, for every r ≥ 2, the class ...

Let V be a finite set with q distinct elements. For a subset C of V n, denote var(C) the variance of the average Hamming distance of C. Let T (n,M; q) andR(n,M; q) denote the minimum and maximum variance of the average Hamming distance of subsets of V n with cardinalityM, respectively. In this paper, we study T (n,M; q) and R(n,M; q) for general q. Using methods from coding theory, we derive up...