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The intersection structure for [Z.sub.2] [Z.sub.4]-additive Hadamard codes is investigated. For any two of these codes [C.sub.1] and [C.sub.2], the Abelian group structure of the intersection [C.sub.1] [intersection] [C.sub.2] is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed, establishing lower and upper bounds for them. Constructions are given of codes whose intersection has any parameters between these bounds. Finally, the intersection problem, i.e., what the possibilities are for the number of codewords in the intersection of two [Z.sub.2] [Z.sub.4]-additive Hadamard codes [C.sub.1] and [C.sub.2] being of the same length, is also studied. Lower and upper bounds for the intersection number are established and, for any value between these bounds, codes with this intersection value are constructed. Index Terms--Additive codes, Hadamard codes, extended perfect codes, perfect codes.

In this paper, new quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension, and the minimum distance, are studied. Using these constructions, new families of quaternary Reed-Muller (RM) codes are built with the peculiarity that after using the Gray map the obtained [Z.sub.4]-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear RM family. To make the duality relationships in the constructed families more evident, the concept of Kronecker inner product is introduced. Index Terms--Additive codes, Kronecker inner product, Plotkin construction, quaternary codes, Reed-Muller (RM) codes, [Z.sub.4]-linear codes.

The intersection problem for [Z.sub.2][Z.sub.4]-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two [Z.sub.2][Z.sub.4]-additive codes [C.sub.1] and [C.sub.2] of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these [Z.sub.2][Z.sub.4]-additive codes [C.sub.1] and [C.sub.2], the abelian group structure of the intersection codes [C.sub.1] N [C.sub.2] is characterized. The parameters of this abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given. Index Terms--Additive codes, extended perfect codes, intersection, perfect codes.

All the possible nonisomorphic additive ([Z.sub.4]-linear and non-[Z.sub.4]-1inear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one. Index Terms--Additive codes, extended perfect codes, Hadamard matrices, Hadamard codes, kernel, rank, [Z.sub.4]-1inear codes.

In this paper, the rank and the dimension of the kernel for (binary) Hadamard codes of length a power of two are studied. In general, it is well-known that the rank of a Hadamard code of length n = [2.sup.t] is a value in {t + 1, ..., n/2}. In the present paper, the range of possible values for the dimension of the kernel is computed and a construction of Hadamard codes of length n = [2.sup.t] for each one of these values is given. Lower and upper bounds for the rank and dimension of the kernel of a Hadamard code of length n = [2.sup.t] are also established. Finally, we construct Hadamard codes for all possible ranks and dimension of kernels between these bounds. Index Terms--Extended perfect codes, Hadamard codes, Hadamard matrices, kernel, rank.

We say that a binary code of length n is additive if it is isomorphic to a subgroup of [Z.sup.[alpha].sub.2] x [Z.sup.[beta].sub.4], where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence [alpha] + 2[beta] = n. In this paper, we prove that any additive extended Preparata-like code always verifies [alpha] = 0, i.e., it is always a [Z.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 [Z.sub.4]-dual of these codes, i.e., the [Z.sub.4]-linear Kerdock-like codes. Index Terms--Additive codes, Kerdock code, kernel, Preparata code, rank, [Z.sub.4]-linear codes.

A binary extended 1-perfect code of length n + 1 = [2.sup.t] is additive if it is a subgroup of [Z.sup.[alpha].sub.2] x [Z.sup.[beta].sub.4]. The punctured code by deleting a [Z.sub.2] coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized in [3] and by using that characterization we compute the possible parameters [alpha], [beta], rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect [Z.sub.4]-linear codes. Index Terms--Additive codes, kernel, perfect codes, rank, [Z.sub.4]-linear codes.

The rank and kernel of 1-perfect additive codes is determined. Additive codes could be seen as translation-invariant propelinear codes and, in this correspondence, a characterization of propelinear codes as codes having a regular subgroup of the full group of isometries of the code is established. A characterization of the automorphism group of a 1-perfect additive code is given and also the cardinality of this group is computed. Finally, an efficiently computable characterization of the Steiner triple systems associated with a 1-perfect binary additive code is also established. Index Terms--Additive codes, automorphism group, kernel, 1-perfect codes, rank.

A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space [F.sup.n], F = {0, 1}, with a group structure. Generally, they are neither Abelian nor translation-invariant codes but they have good algebraic and combinatorial properties. Linear codes and [Z.sub.4]-linear codes can be seen as a subclass of propelinear codes. It is shown here that the subclass of translation-invariant propelinear codes is of type [Mathematical Expression Omitted] where [Q.sub.8] is the non-Abelian quaternion group of eight elements. Exactly, every translation-invariant propelinear code of length n can be seen as a subgroup of [Mathematical Expression Omitted] with [k.sub.1] + 2[k.sub.2] + 4[k.sub.3] = n. For [k.sub.2] = [k.sub.3] = 0 we obtain linear binary codes and for [k.sub.1] = [k.sub.3] = 0 we obtain [Z.sub.4]-linear codes. The class of additive propelinear codes - the Abellan subclass of the translation-invariant propelinear codes - is studied and a family of nonlinear binary perfect codes with a very simply construction and a very simply decoding algorithm is presented. Index Terms - Propelinear codes, translation-invariant propelinear codes, additive codes, perfect codes, 24-linear codes, [Q.sub.8]codes.

The class of completely transitive codes was introduced by Sole 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 proved the nonexistence of completely transitive codes with more than two codewords and error-correcting capability e [is greater than] 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. Index Terms--Completely regular codes, completely transitive codes, permutation groups.

Completely transitive codes were introduced by P. Sole 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. Index Terms: Completely regular codes, completely transitive codes, permutation groups.

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 [Theta]: [F.sup.n] [approaches] [Omega], where [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 always systematic, and we prove their unicity. Therefore, we give a full characterization. Easy coding and decoding algorithms are also presented. Index Terms - Distance-compatible additive codes, perfect codes, translation-invariant propelinear codes.