A ℤ2ℤ4-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2coordinates and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a ℤ2ℤ4-additive cyclic code C, that is, the dimensions of the binary linear codes (Φ(C)) and ker (Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes (Φ(C)) and ker (Φ(C)) are binary images of ℤ2ℤ4-additive codes that we denote by R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code C.

A 762764-additive code C ⊆ ℤ2α× ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of ℤ2ℤ4-additive cyclic codes. We determine all ℤ2ℤ4-additive cyclic codes with odd β whose Gray images are linear binary codes. In this case, it is shown that such binary codes are permutation equivalent (by the Nechaev permutation) to ℤ2-double cyclic codes. Finally, the generator polynomials of these binary codes are given.

A ℤ2ℤ4-additive code C ⊆ ℤ2αx ℤ4βis called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ2and the set of ℤ4coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the ℤ4[x]-module ℤ2[x]/(xα- 1) x ℤ4[x]/(xβ- 1). The parameters of a ℤ2ℤ4-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ℤ2ℤ4-additive cyclic code are determined in terms of the generator polynomials of the code C.

Following [5], given a Z2Z4-additive self-dual code, one can easily extend this code and generate an extended Z2Z4-additive self-dual code with greater length. In this communication we study these constructions and check if properties like separability and code Type are retained or not.

We describe recent results for codes over Z2×Z4giving their connection to binary codes via a natural Gray map. We study Z2Z4self-dual codes and we state the major results concerning these codes. We state several open questions and discuss possible avenues of research.

Quaternary ZRM(r,m) codes were defined so that their binary images, via Gray map, are Reed-Muller codes for some specific values of . In the literature, two different definitions of such codes can be found. They will be denoted ZRM(r,m) and ZRM-(r,m) codes. In this correspondence, we show that both definitions are equivalent exactly for those values of r such that their binary images are Reed-Muller codes. Moreover, we prove that, for all r, these binary images are linear codes in the case of ZRM(r,m), but they are not if we use the definition of ZRM-(r,m). In this last case, we compute the rank and the dimension of the kernel of these codes.

The class of quaternary Reed-Muller codes is introduced as a generalization of Reed-Muller codes to /spl Zopf//sub 4/-linear codes. Properties of this class of codes are established including the rank and kernel of the Gray map image of codes in this class. The class includes all /spl Zopf//sub 4/-linear Kerdock-like and Preparata-like codes.

A binary extended 1-perfect code of length n + 1 = 2/sup t/ is additive if it is a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//. The punctured code by deleting a /spl Zopf//sub 2/ coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized and by using that characterization we compute the possible parameters /spl alpha/, /spl beta/, rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect /spl Zopf//sub 4/-linear codes.

We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.

The class of completely transitive codes was introduced by Sole (1990) as a proper subclass of binary linear completely regular codes. There exist completely transitive codes with error-correcting capabilities e=1, 2, and 3. In a previous correspondence, Borges and Rifa (see ibid., vol.46, no.1, p.279-80, Jan. 2000) proved the nonexistence of completely transitive codes with more than two codewords and error-correcting capability e>4. In this correspondence, we prove the nonexistence for the remaining case, namely, e=4. Therefore, the question of the existence of such codes, depending on their error-correcting capability, is completely solved.

Completely transitive codes were introduced by P. Sole (1990) as a special case of binary linear completely regular codes. The existence of such codes is closely related to the existence of certain permutation groups. The nonexistence of highly transitive permutation groups allows us to prove the nonexistence of completely transitive codes with error-correcting capability greater than 4.

The characterization of perfect single error-correcting codes, or 1 perfect codes, has been an open question for a long time. Recently, Rifa has proved that a binary 1-perfect code can be viewed as a distance-compatible structure in F/sup n/ and a homomorphism /spl theta/:F/sup n//spl rarr//spl Omega/ where /spl Omega/ is a loop (a quasi-group with identity element). In this correspondence, we consider 1-perfect codes that are subgroups of F/sup n/ with a distance-compatible Abelian structure. We compute the set of admissible parameters and give a construction for each case. We also prove that two such codes are different if they have different parameters. The resulting codes are always systematic, and we prove their unicity. Therefore, we give a full characterization. Easy coding and decoding algorithms are also presented.