After several remarks on two-weight irreducible cyclic codes, we introduce a family of projective two-weight cyclic codes and a family of projective two-weight constacyclic codes

We prove the following result on a 2-weight projective cyclic code C over Fq. If q=2 then C is irreducible. If qne2 then either C is irreducible or C is the direct sum of two 1-weight irreducible cyclic codes

We determine all linear cyclic codes over Z/sub 4/ of odd length whose Gray images are linear codes (or, equivalently, whose Nechaev-Gray (1989) image are linear cyclic codes or are linear cyclic codes).

The negashift /spl nu/ of Z/sub 4//sup n/ is defined as the permutation of Z/sub 4//sup n/ such that /spl nu/(a/sub 0/, a/sub 1/, /spl middot//spl middot//spl middot/, a/sub i/, /spl middot//spl middot//spl middot/, a/sub n-1/)=(-a/sub n-1/, a/sub 0/, /spl middot//spl middot//spl middot/, a/sub i/, /spl middot//spl middot//spl middot/, a/sub n-2/) and a negacyclic code of length n over Z/sub 4/ is defined as a subset C of Z/sub 4//sup n/ such that /spl nu/(C)=C. We prove that the Gray image of a linear negacyclic code over Z/sub 4/ of length n is a binary distance invariant (not necessary linear) cyclic code. We also prove that, if n is odd, then every binary code which is the Gray image of a linear cyclic code over Z/sub 4/ of length n is equivalent to a (not necessary linear) cyclic code and this equivalence is explicitely described. This last result explains and generalizes the existence, already known, of versions of Kerdock, Preparata, and others codes as doubly extended cyclic codes. Furthermore, we introduce a family of binary linear cyclic codes which are Gray images of Z/sub 4/ linear negacyclic codes.

Let s and k be integers such that s is a divisor of 2/sup k/-1. Let g(x) be a divisor of x/sup s/-1 over F/sub 2/, and let /spl pi/(x) be a primitive polynomial of degree k over F/sub 2/. We consider the binary cyclic code C of length N=2/sup k/-1 generated by (X/sup N/-1)/g(x)/spl pi/(x). For special cases, we determine the weight distribution of C by using the weights of the cyclic code of length s generated by (x/sup s/-1)/g(x).>

Almost perfect autocorrelation sequences are defined as complex periodic sequences such that all the out-of-phase autocorrelation coefficients are zero except one. The study is restricted to (-1,+1)-sequences. In this case, such sequences exist only if the period n is a multiple of 4. After setting up theoretical results, several sequences are constructed for every period n multiple of 4 in the range 8>

Starting from results on elliptic curves and Kloosterman sums over the finite field GE(2/sup t/), the authors determine the weights of the orthogonals of some binary linear codes; the Melas code of length, the irreducible cyclic binary code of length 2/sup t/+1, and the extended binary Goppa codes defined by polynomials of degree two.>

We give a minimal set consisting of14permutations to decode the(24,12,8)Golay code using a permutation decoding method.