On Generalized Expanded Blaum-Roth Codes
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Expanded Blaum-Roth (EBR) codes consist of n× n arrays such that lines of slopes i, 0≤ i≤ r-1 for 2≤ 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 pτ× (k+r) arrays, where p is prime and τ≥ 1, such that lines of slopes i, 0≤ i≤ r-1, have even parity and every column in the array, when regarded as a polynomial, is a multiple of 1+x^τ. In particular, it was shown that when p is an odd prime number, 2 is primitive in GF(p) and τ = p^j, j≥ 0, the GEBR code consisting of pτ× (p-1)τ arrays is MDS. We extend this result further by proving that GEBR codes consisting of pτ× pτ arrays are MDS if and only if τ = p^j, where 0≤ j and p is any odd prime.
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