Some Specific Features in the Construction of P-Ary Reed-Solomon Codes for an Arbitrary Prime P

Abstract

The Reed-Solomon (RS) codes, proposed in 1960 by Irving Reed and Gustav Solomon as a subset of error-correcting codes, have many current applications. The most significant of which are data recovery in storage systems, including hard drives, minidiscs, CDs, DVDs, Google's GFS, BigTable, and RAID 6, as well as in communication systems such as DSL, WiMAX, DVB, ATSC, and satellite communications. Additionally, RS codes are used as Bar codes in management and advertising systems, such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec Code. Nowadays, RS codes over Galois Fields GF(2m) with base 2 are commonly used in these applications, with the GF(28) field being the most widely used. This allows all 256 values of a byte to be represented as a polynomial with 8 binary coefficients over GF(28). Considering RS codes as cyclic codes in GF(2m) fields, as well as the validity of mathematical dependencies in arbitrary field GF(pm), is a motivation to verify and generalize the idea of generating RS codes in a field with base other prime than 2. As a result, the paper derives the specific features of the construction of Reed-Solomon codes by considering them as a family of codes over any field GF(pm) whose base is a prime p other than 2. The paper also discusses the unique properties of basic arithmetic operations in the arbitrary field GF(pm), which arise from the non-uniqueness of the inverse elements a and -a in a field with base other than 2.