It really is satisfying that this textbook remains to be sufficiently well known to warrant a 3rd version. i've got used the chance to enhance and magnify the publication. while the second one variation used to be ready, merely pages on algebraic geometry codes have been additional. those have now been got rid of and changed via a comparatively lengthy bankruptcy in this topic. even though it remains to be merely an advent, the bankruptcy calls for extra mathematical history of the reader than the rest of this ebook. one of many very attention-grabbing contemporary advancements matters binary codes outlined through the use of codes over the alphabet 7l.4• there's quite a bit curiosity during this zone bankruptcy at the necessities was once further. wisdom of this bankruptcy will enable the reader to review fresh literature on 7l. -codes. four additionally, a few fabric has been further that seemed in my Springer Lec ture Notes 201, yet was once now not integrated in prior versions of this publication, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In bankruptcy 2, a piece on "Coding achieve" ( the engineer's justification for utilizing error-correcting codes) was once additional. For the writer, getting ready this 3rd variation was once a so much welcome go back to arithmetic after seven years of management. For precious discussions at the new fabric, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. a unique be aware of due to R. A. Pellikaan for his suggestions with bankruptcy 10.
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Extra info for Introduction to Coding Theory (Graduate Texts in Mathematics)
Three. 8». We build a code C with codewords zero, 1, the rows of t(S + I + J) and t( -S + I + J). From Theorem 1. three. eight it follows that C is an (n, 2(n + 1), d) code, the place d = t(n - 1) if n == 1 (mod four) and d = t(n - three) if n == three (mod 4). within the case n = nine the code includes the rows of the matrix 000 (4. 1. 1) J P p2 p2 J P J_p2 I J-P J _p2 1 1 1 the place I and J are three by means of three and 000 000 P p2 I J J-P J _ p2 J-P 1 1 1 I 1 1 1 [0 I p= zero zero 1 zero ! ] 4. a few reliable Codes forty eight §4. 2. The Binary Golay Code the main recognized of all (binary) codes is the so-called binary Golay code f§23' There are very many structures of this code, a few of them fairly based and with brief proofs of the homes of this code. we will end up that f§24' the prolonged binary Golay code, is exclusive and deal with a number of buildings. From those it follows that the automorphism staff of the prolonged code is transitive and for that reason f§23 can be designated. We give some thought to the prevalence matrix N of a 2-(11, 6, three) layout. one could exhibit (by hand) that this layout is exclusive. we have now N NT = 31 + 3J. examine N as a matrix with entries in 1F2 • Then NNT = I + J. So N has rank 10 and the one nonzero vector x with xN = zero is 1. The layout houses indicate trivially that the rows of N all have weight 6, and that the sum of any specified rows of N additionally has weight 6. moreover, we all know that the sum of 3 or 4 rows of N is notO. subsequent, permit G be the 1~'by 24 matrix (over 1F2 ) given by means of G := (/12P), the place (4. 2. 1) P:- [I N 1 each row of G has a weight == zero (mod 4). Any rows of G have internal product O. this means that the burden of any linear blend of the rows ofG is == O(mod four) (proof through induction). The observations made approximately N then exhibit linear mix of any variety of rows of G has weight at the very least eight. reflect on the binary code generated through G and get in touch with it f§24' Delete any coordinate to discover a binary [23, 12] code with minimal distance at the very least 7. the gap can't be greater, considering the fact that (3. 1. 6) is happy with e = three, which indicates that during truth this [23, 12,7] code is an ideal code! We denote this code by way of f§23; (as pointed out above, we will end up its area of expertise, justifying the notation). (4. 2. 2) Theorem. The codewords of weight eight in f§24 shape a 5-(24, eight, 1) layout. evidence. by means of a simple counting argument, one could express that the load enumerator of an ideal code containing zero is uniquely made up our minds. actually, we've Ao = A23 = 1, A7 = A16 = 253, As = A 1S = 506, All = A12 = 1288. So, f§24 has 759 phrases of weight eight, no overlapping in additional than 4 positions. zero for this reason, those phrases jointly disguise 759· (~) = CZs4 ) fivetuples. (4. 2. three) Theorem. If C is a binary code of size 24, with distance eight, and if zero E C, then C "is such as f§24' ICI = 212, minimal facts. (i) The tricky a part of the evidence is to teach that C has to be a linear code. to work out this, notice that deleting any coordinate produces a code C' of size 23 and distance 7 with IC'I = 212. So, this code is ideal and its weight 49 §4.