Book Review: Algebraic-geometric codes
Jacobus H. Lint · Bulletin of the American Mathematical Society · 1992
In the theory of error-correcting codes a code C is a subset of Q" where Q is a finite set called the alphabet.The elements of C are called codewords (or vectors) and n is called the wordlength.In practice, the codewords are messages that are sent over a so-called "noisy" channel to a receiver.The channel has the effect that if a codeword c is sent, the received word r may differ from c in a number of places.We say that errors occur in the received message.In the set Q" a distance function is introduced by d(x, y) := \{i : 1 < i < n, x¡ ¿ y¡}\.The minimum distance d of the code C is defined by d := min{d(\, y) : x e C, yeC, x^y}.If d -2e + 1, then the code C is called an e-error-correcting code because if a received word r has distance < e to the transmitted message c, the receiver can "correct" the errors and retrieve the word c, since the distance from r to all other codewords is larger than e .Shannon's paper (1948) on the mathematical theory of communication [8] marks the beginning of coding theory.Since then, most of the theory has been concerned with so-called linear codes.For the alphabet Q one chooses a finite field ¥q and the code C is a linear subspace of F£ .If C has dimension k , then C is called an [n, k] code.The easiest situation is that of a systematic code C, where the first k coordinates Ci , ... , cfc of codewords c take on all qk possible values and the code is obtained by a mapping from F*j, to F£ that adjoins "redundant" symbols ck+l, ... , cn .The efficiency of C for transmission of information is measured by the ratio k/n , which is called the information rate of C.If C is a <7-ary [n, k] code with minimum distance d, then two distinct codewords cannot be identical on the first n-(d -\) positions and, therefore, the number of codewords cannot exceed q"-d+l .This yields what is known as the Singleton bound d<n-k+ 1.