Partial permutation decoding for MacDonald codes

Abstract

We show how to find s-PD-sets of the minimal size $s+1$ for the $\left[ \frac{q^n-q^u}{q-1},n,q^{n-1}-q^{u-1}\right] _q$ MacDonald q-ary codes $C_{n,u}(q)$ where $n \ge 3$ and $1 \le u \le n-1$. The construction of [6] can be used and gives s-PD-sets for s up to the bound $\lfloor \frac{q^{n-u}-1}{(n-u)(q-1)} \rfloor -1$, of effective use for u small; for $u \ge \lfloor \frac{n}{2} \rfloor$ an alternative construction is given that applies up to a bound that depends on the maximum size of a set of vectors in $V_u(\mathbb {F}_q)$ with each pair of vectors distance at least 3 apart.

Appendix

In Table 1 for $C_{n,u}(q), q$ is a prime power, n, u as before, $\ell$ is the length of the code, mw is the minimum weight, t is the error-correcting capability, gb is the Gordon-Schönheim bound for the size of the PD-set for full error correction, $s_1=f_{n-u}$ (for the first construction), $s_2$ the computationally found value of s for the second construction when $u \ge 3$. Thus the size of the set is $s_1+1, s_2+1$, respectively.

Table 1 s-PD-sets of size $s+1$ for $C_{n,u}(q)$