Half-cyclic, dihedral and half-dihedral codes

Abstract

This note discusses, in elementary terms, linear codes over $\mathbb {Z}_2$ which are closed under 2-step cyclic shifts, and classifies them in terms of special linear combinations of polynomials. Codes which are preserved under order-reversing automorphisms are also discussed, and a classification result in terms of special linear combinations of polynomials is obtained.

Appendix

Tables 1 and 2 display all half-cyclic code of length 8 or less. One of each associate pair of generating pairs and only one pair of generating pairs for a code and its reverse are given. The triple of parameters [n, k, m] denotes, as usual, the length, the dimension and the minimal distance of the code, while [a, b] is the pair of the degrees of the generators. The next columns present canonical generators u and v without their trailing zeroes. The ‘type’ column shows

$$\begin{aligned} \begin{array}{ll} \text {Cyc}&{}\,\,\,\text {cyclic}\\ \text {Dih}&{}\,\,\, \text {dihedral}\\ \text {HC}&{}\,\,\,\text {half-cyclic}\\ \text {HDiEvn}&{}\,\,\, \text {even half-dihedral}\\ \text {HDiOdd}&{}\,\,\, \text {odd half-dihedral}\\ \text {--DG}&{}\,\,\,\text {degenerate} \end{array} \end{aligned}$$

A more complete listing of codes up to length 14 (some 540 in all) is available on the website http://euler.doa.fmph.uniba.sk/half-dihedral-codes.html

Table 1 Half-cyclic codes of length up to 6

Table 2 Half-cyclic codes of length 8

The following example illustrates some of these possibilities:

Example 3.8

Consider the six codes of length 10 given in Table 3 (in the form of entries from Table 1).

Table 3 Some related codes of length 10

In general coding theory, two codes are considered equivalent if some permutation of coordinates applied uniformly across one code produces the other. For a half-cyclic code, most permutations of coordinates applied to it do not yield a half-cyclic code. There are, however, two permutations from the full symmetric group ${\mathbb S}_n$ under which the collection of half-cyclic codes is obviously closed, namely, $\varphi = (1,3,5,\dots ,n-1)$, and $\psi = (0,2,4,\dots ,n-2)$. For any half-cyclic code C, both $C\varphi$ and $C \psi$ are also half-cyclic. The generators for $C\varphi$ and $C \psi$ generally differ from those of C, and they are not always the images of the generators of C under $\varphi$ or $\psi$. The basic parameters [n, k, m], though, remain unchanged. The type may change quite freely.

If, for any half-cyclic code H generated by u and v, we let $\bar{H}$ denote the code generated by $\bar{u}$ and $\bar{v}$, the reader may easily check that $A\varphi = \bar{B}$, $B\varphi = \bar{E}$, $C\varphi = \bar{C}$, $D\varphi = \bar{F}$, $E\varphi = \bar{E}R$, and $F\varphi = DR$.

Final Remark While this paper has been concerned exclusively with codes over $\mathbb {Z}_2$, almost all our definitions, techniques and results would apply to codes over other fields as well.