By Jacobus H. van Lint (auth.)
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Extra resources for Coding Theory
N, -', 39 j ~ O""'~2-' 1s transmitted as the sequence ~ :- (co'c" ••• ,c ck :& Rij if k 2 i (mod n,) and k ~ j (mod ~). :11:12 ~ where By the Chinese remainder theorem £. Using this cyclic this is a one-one correspondence between the elements of A and order1:tg a burst error in _I) will be distributed among the rows and columns of A. It would be very useful if a not too complicated decoding algorithm for product codes cOl:ld be f:JUnd which actually corrects t:-te error patter:ts that one knows can be corrected.
1 + 1. 1) Xl + x2n+l If the received IoIOrd c O. is 'f.. = :!. ~ (i = 2, ... ,2n+ 1). If more than n of the expressions Yl + Y1 is , this can be explained by either less than n . errors among which e, likely. n r. errors and e 1 " O. he )'1 + Yi de~ides whether e 1 ber of votes exceeds the threshold n then we set e 1 = ~he c former is m::>re C or 1. I! t;,e num- 1. 2) DEFINITION: Let V be a linear code with bl::>ck length n. ln; in at Ill::>st one ::>f the c::>ef:icient. now demonGtra1;e how to usc guch a set of parity-check equations.
He )'1 + Yi de~ides whether e 1 ber of votes exceeds the threshold n then we set e 1 = ~he c former is m::>re C or 1. I! t;,e num- 1. 2) DEFINITION: Let V be a linear code with bl::>ck length n. ln; in at Ill::>st one ::>f the c::>ef:icient. now demonGtra1;e how to usc guch a set of parity-check equations. sider the dual of the (15,11) Hamming code. t the 3 columns SUlll to zer::>. la::' code with three terms. 3) r XlXI 1 Among t;,ese the set of 7 equations: x2 + x 3 ' x4 + x j ' .. oer 1. on3: The code under "onsideration has minim'JlIl distance 8.