Abstract We construct a family of cyclic completely regular codes (CRCs) of length $$n=2^m-1$$ n = 2 m - 1 and compute their intersection arrays. These codes, denoted $$C_{1,5}$$ C 1 , 5 , are generated by the product $$m_1(x)m_5(x)$$ m 1 ( x ) m 5 ( x ) , where $$m_i(x)$$ m i ( x ) is the minimal polynomial of $$\alpha ^i$$ α i , and $$\alpha $$ α is a primitive element of the finite field $$\mathbb {F}_{2^m}$$ F 2 m . We consider two main cases: odd m and $$m \equiv 2 \pmod {4}$$ m ≡ 2 ( mod 4 ) . For odd m , these codes are known to be completely regular with covering radius $$\rho =3$$ ρ = 3 and minimum distance $$d=5$$ d = 5 . We prove that, for any m , the codes $$C_{1,3}$$ C 1 , 3 and $$C_{1,5}$$ C 1 , 5 are non-equivalent despite sharing the same parameters and intersection array. For $$m \equiv 2 \pmod {4}$$ m ≡

It is known that$\mathbb {Z}_{p^{s}}$-linear codes, which are the Gray map image of$\mathbb {Z}_{p^{s}}$-additive codes (linear codes over$\mathbb {Z}_{p^{s}}$), are systematic and a systematic encoding has been found. This makes$\mathbb {Z}_{p^{s}}$-linear codes suitable to apply the permutation decoding method, based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. Some constructions of r-PD-sets of minimum size$r+1$for$\mathbb {Z}_{p^{s}}$-linear generalized Hadamard codes of type$(n;t_{1}, {\dots },t_{s})$are known. In this paper, for these codes, we present new constructions of r-PD-sets of size$r+1$, which are suitable for all parameters$t_{1}, {\dots },t_{s}$. These allow us to obtain new r-PD-sets for values of r closer to the theoretical upper bound, improving previous known results.

A subset of a vector space$\mathbb {F}_{q}^{n}$is additive if it is a linear space over the field$\mathbb {F}_{p}$, where$q=p^{e}$,$p$prime, and$e>1$. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case$e=2$, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

A subset of a vector space Fqnis K-additive if it is a linear space over the subfield K ⊆ Fq. Let q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are Fp-additive are established. For specific ranks and dimensions of the kernel within these bounds, Fp-additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an Fp-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq, q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.

This paper deals with Hadamard Z2Z4Q8-codes, which are binary codes after a Gray map from a subgroup of direct products of Z2, Z4, and Q8, where Q8is the quaternionic group. In a previous work, these codes were classified in five shapes. In this paper, we analyze the allowable range of values for the rank and dimension of the kernel, which depends on the particular shape of the code. We show that all these codes can be represented in a standard form, from a set of generators, which can help in understanding the characteristics of each shape. The main results we present are the characterization of Hadamard Z2Z4Q8-codes as a quotient of a semidirect product of Z2Z4-linear codes and the construction of Hadamard Z2Z4Q8-codes with each allowable pair of values for the rank and dimension of the kernel.

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2m- 1 for small m (m ≤ 22), is given. For m ∈{5, 7,11,13,17,19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is proposed. This classification for APN monomial power permutations coincides with the CCZ-equivalence, at least up to m ≤ 17.

A$\BBZ_{2}\BBZ_{4}Q_{8}$-code is the binary image, after a Gray map, of a subgroup of$\BBZ_{2}^{k_{1}}\times\BBZ_{4}^{k_{2}}\times Q_{8}^{k_{3}}$, where$Q_{8}$is the quaternion group on eight elements. Such$\BBZ_{2}\BBZ_{4}Q_{8}$-codes are translation invariant propelinear codes as are the well known$\BBZ_{4}$-linear or$\BBZ_{2}\BBZ_{4}$-linear codes. In this paper, we show that there exist “pure”$\BBZ_{2}\BBZ_{4}Q_{8}$-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the$\BBZ_{2}\BBZ_{4}Q_{8}$-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of$\BBZ_{2}\BBZ_{4}Q_{8}$-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard.

In this paper, we consider completely regular codes, obtained from perfect (Hamming) codes by lifting the ground field. More exactly, for a given Hamming code$C$of length$n=(q^m-1)/(q-1)$over${\BBF}_q$with a parity check matrix$H_m$, we define a new linear code$C_{(m,r)}$of length$n$over${\BBF}_{q^r},$$r \geq 2$, with this parity check matrix$H_m$. The resulting code$C_{(m,r)}$is completely regular with covering radius$\rho = \min\{r,m\}$. We compute the intersection numbers of such codes and we prove that Hamming codes are the only codes that, after lifting the ground field, result in completely regular codes. Finally, we also prove that extended perfect (Hamming) codes, for the case when extension increases their minimum distance, are the only codes that, after lifting the ground field, result in uniformly packed (in the wide sense) codes.

Product perfect codes have been proven to enhance the performance of the F5 steganographic method, whereas perfect Z2Z4-linear codes have been recently introduced as an efficient way to embed data, conforming to the ±1-steganography. In this paper, we present two steganographic methods. On the one hand, a generalization of product perfect codes is made. On the other hand, this generalization is applied to perfect Z2Z4-linear codes. Finally, the performance of the proposed methods is evaluated and compared with those of the aforementioned schemes.

For any integer ¿ ¿ 1 and for any prime power q , an explicit construction of an infinite family of completely regular (and completely transitive) q-ary codes withd= 3 and with covering radius ¿ is given. The intersection array is also computed. Under the same conditions, the explicit construction of an infinite family of q-ary uniformly packed codes (in the wide sense) with covering radius ¿, which are not completely regular, is also given. In both constructions, the Kronecker product is the basic tool that has been used.

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 Z4-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.

The intersection structure forZ2Z4-additive Hadamard codes is investigated. For any two of these codesC1andC2, the Abelian group structure of the intersectionC1capC2is 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.

The intersection problem for Z2Z4-additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z2Z4-additive codes C1and C2of 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 Z2Z4-additive codes C1and C2, the abelian group structure of the intersection codes C1cap C2is 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.

All the possible nonisomorphic additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.

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.

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 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 isometrics 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.

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.

Binary 1-perfect codes which give rise to partitions of the n-cube are presented. The 1-perfect partitions are characterized as homomorphic images of simple algebraic structures on F n and are constructed starting from a particular case of a structure defined in F n . A special property (so-called well-ordering) of STS(n) is given in such a way that for this kind of STS it is possible to define the algebraic structure we need in F n and to construct 1-perfect partitions of the n-cube. These 1-perfect partitions give us a kind of 1-perfect code for which it is easy to do the coding and decoding. Furthermore, there exists a syndrome which allows us to perform error correction. We present systematic codes of length n=15 and we give examples of how to do the coding, decoding, and error correction.

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.

A class of binary group codes is investigated. These codes are the propelinear codes, defined over the Hamming metric space F/sup m/, 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 Z/sub 2//sup k1//spl oplus/Z/sub 4//sup k2//spl oplus/Q/sub 8/(k3) 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 Z/sub 2//sup k1//spl oplus/Z/sub 4//sup k2//spl oplus/Q/sub 8//sup k3/ with k/sub 1/+2k/sub 2/+4k/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 Abelian 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.

Block codes which allow error correction in a two-dimensional QAM signal space are given. The properties of these codes are used to demodulate QAM signals in a differentially coherent detection scheme on a noisy channel. The block codes presented are not group codes; however, their components belong to a group G/sub n/ or G'/sub n/ which is constructed starting from the Gaussian integers G=Z[i] modulo a nonprime ideal (2/sup n/+2/sup n/i) and taking on it the multiplicative group of units. The authors classify and factor these multiplicative groups and use them to construct the block codes which become optimal and efficient from the point of view of transmission rate and decoding.>