Nearly MDS expander codes with reduced alphabet size

Recently, Roth and Skachek proposed two methods for constructing nearly maximum-distance separable (MDS) expander codes. We show that through the simple modification of using mixed-alphabet codes derived from MDS codes as constituent codes in their code designs, one can obtain nearly MDS codes of significantly smaller alphabet size, albeit at t he expense of a (very slight) reduction in code rate. ) , respectively, where αR is dependent on R but may nevertheless be upper bounded by a universal constant independent of R. This correspondence furthers the pursuit of reducing the alphabet size of nearly MDS expander codes. We will show that by using mixed-alphabet codes derived from single- alphabet MDS codes as constituent codes in the constructions by Roth and Skachek, one obtains (i) linear-time decodable nearly MDS codes of rate at least R ǫ where R > ǫ + 1/2 and with an alphabet size reduced by a factor of about 1 ǫ in the exponent, and (ii) linear-time encodable and decodable nearly MDS codes of rate at least R ǫ where R = 1O (ǫ) and with an alphabet size reduced by a factor of about 1O (ǫ) in the exponent. We point out that in both cases, the reduction in alphabet size does not come for free and is, in fact, at the expense of a reduction in rate. Nevertheless, we will show that a significant improvement in alphabet size may be achieved at a price of a very small reduction in rate.