Coding Theory, 2nd Edition by Jacobus H. van Lint (auth.)

By Jacobus H. van Lint (auth.)

These lecture notes are the contents of a two-term direction given through me in the course of the 1970-1971 educational yr as Morgan Ward traveling professor on the California Institute of expertise. the scholars who took the path have been arithmetic seniors and graduate scholars. hence an intensive wisdom of algebra. (a. o. linear algebra, thought of finite fields, characters of abelian teams) and likewise likelihood idea have been assumed. After introducing coding idea and linear codes those notes quandary themes typically from algebraic coding idea. the sensible part of the topic, e. g. circuitry, isn't really incorporated. a few subject matters which one want to comprise 1n a path for college kids of arithmetic equivalent to bounds at the details cost of codes and plenty of connections among combinatorial arithmetic and coding thought couldn't be handled as a result of loss of time. For an extension of the direction right into a 3rd time period those issues may were selected. even supposing the fabric for this path got here from many resources there are 3 which contributed seriously and which have been used as advised studying fabric for the scholars. those are W. W. Peterson's Error-Correcting Codes «(15]), E. R. Berlekamp's Algebraic Coding idea «(5]) and several other of the AFCRL-reports by way of E. F. Assmus, H. F. Mattson and R. Turyn ([2], (3), [4] a. o. ). For numerous fruitful discussions i need to thank R. J. McEliece.

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The smallest ideal which contains 2 V, and V2 • Notice that V, n V2 1s also an ideal in R. (i) V, n V2 is generated by the least common multiple of g, (x) and g2(x), (ii) V, + V2 is generated by the greatest common divisor of g,(x) and ~(x). We leave the proof to the reader as an exercise. (x». J From (3. 15) it follows that an ideal V in R is the "sum" of the minimal ideals contained in V. To conclude this introduction we treat a simple example namely n We have x 7 _ , = 2 (X_l)(x 3+x +1)(x 3+x+l).

Is called a cycle of length e. d. (x -l)/f i 2) of M~ has length e • i since every nonzero code (X). Now suppose we have cycle representatives for the codes M~ and ~ M:. J a(x) E M~1 b(x) E Mjl h := (per(~), per<:~», H := [per(~), per(~)]. a,' a'k T ! + T a. a,' ~' then T ! - T ~ =T £. g. let If TOa+ TSb = but since the vord on the left-hand side is in M- and the one on the right-hand side is in i M:J they must both be o. - Therefore TO! + T$E. per(~) different values. These are cod-e words in M~+ Mj.

3) THEOREM: If ~ is a primitive n-th root of unity in GF(q) ~ q = pk (k is the mult. order of p modn) then the set V := (£(s) := (Tr(~), Tr(~~), ••• , Tr(~~n-l»1 ~ e GF(q)} is an irreducible cyclic code (of length n and dimension k). 2) (c) and (d) V is a linear code. Next we note that £(~S-l) is a cyclic shift of £(~). Therefore V is a cyclic code. xk O since S is in no subfield of GF(q). O the code V. e. we have a parity check equation for Since h(x) is irreducible we see that xkh(x-') 1s the check.

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