On Generalized Expanded Blaum-Roth Codes

Expanded Blaum-Roth (EBR) codes consist of $n\times n$ arrays such that lines of slopes $i$, $0\leq i\leq r-1$ for $2\leq r<n$, as well as vertical lines, have even parity. The codes are MDS with respect to columns, i.e., they can recover any $r$ erased columns, if and only if $n$ is a prime number. Recently a generalization of EBR codes, called generalized expanded Blaum-Roth (GEBR) codes, was presented. GEBR codes consist of $p\tau\times (k+r)$ arrays, where $p$ is prime and $\tau\geq 1$, such that lines of slopes $i$, $0\leq i\leq r-1$, have even parity and every column in the array, when regarded as a polynomial, is a multiple of $1+x^{\tau}$. In particular, it was shown that when $p$ is an odd prime number, 2 is primitive in $GF(p)$ and $\tau = p^j$, $j\geq 0$, the GEBR code consisting of $p\tau\times (p-1)\tau$ arrays is MDS. We extend this result further by proving that GEBR codes consisting of $p\tau\times p\tau$ arrays are MDS if and only if $\tau = p^j$, where $0\leq j$ and $p$ is any odd prime.