We show that Han’s bijection when restricted to permutations can be carried out in terms of the major code and inversion code. In other words, it maps a permutation π with a major code (s1, s2, . . . , sn) to a permutation σ with an inversion code (s1, s2, . . . , sn). We also show that the fixed points of Han’s map can be characterized by the strong fixed points of Foata’s second fundamental t...

We show that the state complexity profile of a convolutional code is the same as that of the reciprocal of the dual code of in case that minimal encoders for both codes are used. Then, we propose an optimum permutation for any given ( 1) binary convolutional code that will yield an equivalent code with the lowest state complexity. With this permutation, we are able to find many ( 1) binary conv...

An (n, d)-permutation code of size s is a subset C of Sn with s elements such that the Hamming distance dH between any two distinct elements of C is at least equal to d. In this paper, we give new upper bounds for the maximal size μ(n, d) of an (n, d)-permutation code of degree n with 11 6 n 6 14. In order to obtain these bounds, we use the structure of association scheme of the permutation gro...

Permutation decoding is a technique, introduced in [3] by MacWilliams, that strongly depends on the existence of special subsets, called PD-sets, of the permutation automorphism group PAut(C) of a linear code C. In [2], it is shown how to find s-PD-sets of minimum size s + 1 for partial permutation decoding for the binary simplex code Sm of length 2 − 1, for all m ≥ 4 and 1 < s ≤ ⌊ 2−m−1 m ⌋ . ...

We give the first Gray code for the set of n-length permutations with a given number of cycles. In this code, each permutation is transformed into its successor by a product with a cycle of length three, which is optimal. If we represent each permutation by its transposition array then the obtained list still remains a Gray code and this allows us to construct a constant amortized time (CAT) al...

We show that Han’s bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation π with a cyclic major code (s1, s2, . . . , sn) to a permutation σ with a cyclic inversion code (s1, s2, . . . , sn). We also show that the fixed points of Han’s map can be characterized by the strong fixed points ...

Given a linear code C, the fundamental problem of trellis decoding is to find a coordinate permutation of C that yields a code C ′ whose minimal trellis has the least state-complexity among all codes obtainable by permuting the coordinates of C. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of finding such a coordinate permutation is NP-hard, thus ...

Consider the symmetric group Sn with the Hamming metric. A permutation code on n symbols is a subset C ⊆ Sn. If C has minimum distance > n − 1, then |C| 6 n2 − n. Equality can be reached if and only if a projective plane of order n exists. Call C embeddable if it is contained in a permutation code of minimum distance n−1 and cardinality n2−n. Let δ = δ(C) = n2−n−|C| be the deficiency of the per...