Generalized MacWilliams identities on SKE weights for linear codes over ring Zq

Abstract

In Coding Theory/ Error-Correcting codes, the concept of distance, or of metric, is used as a measure of degree of dissimilarity between two words of equal length, which may be transmitted/ received in communication systems, or stored/ retrieved in digital storage devices. For the Euclidean-type, one of the types of distances, distance (or sometime, the square of the distance) between two words a = (a₀, a₁, …, a_n) and b = (b₀, b₁, …, b_n) is the sum of the squares of the values—under the particular metric/distance of Euclidean-type—of differences (a_i  b_i) in the corresponding tuples of the two words a and b, for i = 1, 2, …, n. Euclidean-type distance is (i) regarded as the most relevant measure of efficiency for symmetric PSK-codes, (ii) used in wireless LAN standard IEEE 802.11b-1999, and (iii) extensively applied in analysis of convolution codes and Trellis codes.

On the other hand, MacWilliams Identities provide a mechanism for deriving properties of large codes from corresponding properties of (generally very) small codes. The identities, which relate weight-enumerators of a code and its dual code, were first derived in 1963 by MacWilliams¹ for linear codes over finite fields for Hamming metric. MacWilliams-type identities for Sharma-Kaushik metrics (SK-metrics) are discussed².

In this paper, we investigate the more general case of the possible MacWilliams-type identities for Sharma-Kaushik Euclidean distance (SKE distance), a new concept to be defined. These investigations generalize the MacWilliams identities for Euclidean distance discussed in^2-3. The results in the investigation have the potential for improving (i) the wireless LAN standard IEEE 802.11b-1999 (ii) functioning of MANET, VANET & other networks.