Repeated-root cyclic codes

In the theory of cyclic codes, it is common practice to require that (n,q)= 1, where n is the word length and Fq is the alphabet. This ensures that the generator g ( x ) of the cyclic code has no multiple zeros ( = repeated mots). Furthermore it makes it possible to use an idempotent element as generator. However, much of the theory also goes through without the restriction on n and q. Recently, the author was asked whether dropping the restriction could produce any good d e s or that they would always be bad (in some sense), in which case making the restriction right after the definition, as most authors do, would he justified. This question led to the results below. We shall show that a binary cyclic code of length 2 n ( n odd) can be obtained from two cyclic codes of length n by the well known lulu + V I construction. This leads to an infinite sequence of optimal cyclic d e s with distance 4. Furthermore, it shows that for these codes low complexity decoding methods can be used. The structure theorem generalizes to other characteristics and to other lengths. Independently, Castagnoli ef al. have studied the same question. Some of their results are similar to these results, but their methods are different. Some comparisons of the methods using earlier examples are also given. I n h T e r m -Binary cyclic codes of even length, shortened Hamming codes. I. BINARY CYCLIC ODES OF LENGTH 2n (n ODD) Let n be odd and x" 1 = f l ( x ) f 2 ( x ) . . . f , ( x ) the factorization of X" 1 into irreducible factors in IF,[x]. We define g l ( x ) : = f l ( x ) . . . f , ( x ) , g2(x):=fk+l(x)...fr(x), where k < 1 < t. Let r , := deg g , , r , := deg g ,g , . Let C , be the cyclic code of length n and dimension n rl with generator g , (x ) , and let C , be the cyclic code of length n and dimension n rz with generator g , (x )g , (x ) , and let d , be the minimum distance of Ci ( i = 1,2). Clearly d , 2 d , . We are interested in the cyclic code C of length 2n and dimension 2n rl r , with generator g ( x ) := g?(x )g , (x ) . We claim that this code has the following structure: Let a = (ao ,a l ; a , u , , ) E C , and c =(c0,c1;. ., c , , ) E C,. Define b := a + c. Since n is odd, we can define words that belong to C by and in this way we find all words of C. The last assertion is a consequence of dimension arguments. We prove the first assertion as follows. Write , b,, 2 , a, 1 9 bo 7 a 1 7 * . * 9 a, z 9 bn 1 ) 9 w : = ( a , , b , , a , , . . . a(.) = a,+ a , x + . .. + a , , X " l = ( a , + a , x 2 + . . . + a , _ , x " l ) + x ( a , + . . . + a , -2x" -3 ) = a,( x 2 ) + xu,( XZ) , Manuscript received December 19, 1989. The author is with Philips Research Laboratories, P.O. Box 80000-5600 JA, Eindhoven, The Netherlands and the Department of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Netherlands. IEEE Log Number 9041643. and analogously for c ( x ) and b(x) . We then have the following two (equivalent) representations for the polynomial w ( x ) wrresponding to the codeword w : w( x ) = {a,( x ' ) + X"+lU, ( 2)) +{ xb,( x ' ) + X"b,( 2))