Glossary

• characteristic: Base prime of the Galois Field, ie $GF(2^4)$ has characteristic 2.
• $q$: Alphabet size of a Galois Field, for characteristic 2 fields this is a power of 2.
• $n$: Block length of a Reed-Solomon code measured in symbols, at most $q-1$ for BCH view encoding.
• $k$: Message length of a Reed-Solomon code measured in symbols.
• $d$: Distance, ie number of symbols that differ, for Reed-Solomon $d=n-k+1$
• $t$: Number of errors that a Reed-Solomon code can correct $t=\lfloor\frac{n-k}{2}\rfloor$.
• $α$: Primitive/generator element whose exponentiation can reach all field values excepting 0, for characteristic 2 fields this is almost always chosen to be 2.
• $m(x)$: The message polynomial, consists of $k$ terms, some more generalized explanations may use $p(x)$ instead, however I'm not entirely clear on the nuance of the difference here.
• $g(x)$: The generator polynomial, in the BCH view of Reed-Solomon codes, is constructed as a polynomial containing roots that correspond to t consecutive powers of $α$. In the systematic encoding, the message polynomial is divided by it and the remainder is the check symbols that get appended to get the encoded message.
• $c$: Aka FCR or first consecutive root. The power $α$ that is the lowest out of the consecutive roots used to create the generator polynomial.
• $s(x)$: The encoded codeword polynomial.
• $e(x)$: The errata magnitude polynomial.
• $r(x)$: The received polynomial. $r(x)=s(x)+e(x)$
• $S(x)$: The syndrome polynomial. The results of evaluating the received polynomial $r(x)$ at the roots introduced during the encoding process, with the FCR in the order zero position and consecutive roots in increasing positions.