The b-Symbol Hamming Weight Spectra of Quaternary Kerdock Codes and Related Codes

Abstract

The symbol-pair coding theory was put forward by Cassuto and Blaum [IEEE TIT, 2011] to be applicable in high-density storage situations. Yaakobi et al. [IEEE TIT, 2016] extended the concept of symbol-pair metric to b-symbol metric when $b\geq 2$ . The extensive research on the b-symbol Hamming weight spectra of cyclic codes has been centered on the case where the alphabet is a finite field. The case of cyclic codes over ${\mathbb {Z}}_{4}$ , an extremely important class of codes, has been overlooked for a long time in the exploration of the b-symbol Hamming weight spectra. In this paper, we study the b-symbol Hamming weight spectra of the shortened Kerdock codes ${\mathcal {K}}_{m}^{-}$ and the Kerdock codes ${\mathcal {K}}_{m}$ over ${\mathbb {Z}}_{4}$ . The formulas for calculating the symbol-pair Hamming weight of the codewords in ${\mathcal {K}}_{m}^{-}$ and ${\mathcal {K}}_{m}$ are given, and their values hinge on the trace-like function values of specific elements in the Teichmüller set. In particular, we present a class of ${\mathbb {Z}}_{4}$ -cyclic codes with three non-zero b-symbol Hamming weights. As by-products, the b-symbol Hamming weight hierarchies of the Preparata codes and the Goethals codes are provided.