These lecture notes are the contents of a two-term course given by me during the 1970-1971 academic year as Morgan Ward visiting professor at the California Institute of Technology. The students who took the course were mathematics seniors and graduate students. Therefore a thorough knowledge of algebra. (a. o. linear algebra, theory of finite fields, characters of abelian groups) and also probability theory were assumed. After introducing coding theory and linear codes these notes concern topics mostly from algebraic coding theory. The practical side of the subject, e. g. circuitry, is not included. Some topics which one would like to include 1n a course for students of mathematics such as bounds on the information rate of codes and many connections between combinatorial mathematics and coding theory could not be treated due to lack of time. For an extension of the course into a third term these two topics would have been chosen. Although the material for this course came from many sources there are three which contributed heavily and which were used as suggested reading material for the students. These are W. W. Peterson's Error-Correcting Codes «(15]), E. R. Berlekamp's Algebraic Coding Theory «(5]) and several of the AFCRL-reports by E. F. Assmus, H. F. Mattson and R. Turyn (, (3),  a. o. ). For several fruitful discussions I would like to thank R. J. McEliece.
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algorithm basis vector BCH code binary cyclic code binary Golay code bits block length burst error called channel check polynomial code is perfect code of block code of length code over GF(q code word coefficients consider correcting code corresponding coset coset leader cycles cyclic code cyclic shift define DEFINITION denote dimension dual code element of GF(2 error pattern error probability example expected number extended code Hamming code Hence idempotent integer LEMMA linear code linear subspace minimum distance minimum weight multiple n a 1 mod number of errors order RM code orthogonal parity check matrix parity-check equations perfect codes perfect e-error-correcting code perfect single-error-correcting code permutation positions primitive element primitive n-th root problem Proof prove quadratic residue r-th order RM remark root of unity rows of G Section sequence subcode syndrome ternary Golay code vector space weight enumerator words of weight zeros