Codes from incidence matrices and line graphs of Paley graphs
Dina Ghinelli Jennifer D. Key
We examine the $p$-ary codes from incidence matrices of Paley graphs $P(q)$ where $q\equiv 1$(mod $4$) is a prime power, and show that the codes are $[\frac{q(q-1)}{4},q-1,\frac{q-1}{2}]$2 or $[\frac{q(q-1)}{4},q,\frac{q-1}{2}]$p for $p$ odd. By finding PD-sets we show that for $q > 9$ the $p$-ary codes, for any $p$, can be used for permutation decoding for full error-correction. The binary code from the line graph of $P(q)$ is shown to be the same as the binary code from an incidence matrix for $P(q)$.
keywords: codes permutation decoding. Paley graphs
Partial permutation decoding for simplex codes
Washiela Fish Jennifer D. Key Eric Mwambene
We show how to find $s$-PD-sets of size $s+1$ that satisfy the Gordon-Schönheim bound for partial permutation decoding for the binary simplex codes $\mathcal S_n(\mathbb F_2)$ for all $n \geq 4$, and for all values of $s$ up to $\left\lfloor\frac{2^n-1}{n}\right\rfloor -1$. The construction also applies to the $q$-ary simplex codes $\mathcal S_n(\mathbb F_q)$ for $q>2$, and to $s$-antiblocking information systems of size $s+1$, for $s$ up to $\left\lfloor\frac{(q^n-1)/(q-1)}{n}\right\rfloor -1$ for all $q$.
keywords: permutation decoding simplex codes antiblocking decoding. Hamming codes
Binary codes from reflexive uniform subset graphs on $3$-sets
Washiela Fish Jennifer D. Key Eric Mwambene
We examine the binary codes $C_2(A_i+I)$ from matrices $A_i+I$ where $A_i$ is an adjacency matrix of a uniform subset graph $\Gamma(n,3,i)$ of $3$-subsets of a set of size $n$ with adjacency defined by subsets meeting in $i$ elements of $\Omega$, where $0 \le i \le 2$. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the $C_2(A_i+I)$ are also examined. We obtain partial PD-sets for some of the codes, for permutation decoding.
keywords: permutation decoding. codes Uniform subset graphs
Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$
Jennifer D. Key Washiela Fish Eric Mwambene
We examine the $p$-ary codes, for any prime $p$, that can be obtained from incidence matrices and line graphs of the Hamming graphs, $H^k(n,m)$, for $k \geq 2$. For $m=2$, we obtain the main parameters of the codes from the incidence matrices, including the minimum weight and the nature of the minimum words. We show that all the codes can be used for full permutation decoding.
keywords: Hamming graphs codes permutation decoding.

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