Given a 1-perfect code C, the group of symmetries of C, Sym(C)={π ∈ Sn | π(C)=C}, is a subgroup of the group of automorphisms of C. In this paper, we focus on symmetries of order 2, i.e., involutions. Let Inv(C) ⊆ Sym(C) be the set of involutions of C. An additive code C of type (α,β) is a subgroup of \BBZ2α×\BBZ4β. For additive 1-perfect codes C of length 2t-1, we establish that any involution π ∈ Inv(C) must have at least 2k-1 fixed points where k ≥ [t/2].

Given a 1-perfect codeC, the group of symmetries ofC,Sym(C)={π ∈Sn| π(C)=C} , is a subgroup of the group of automorphisms ofC. In this paper, we focus on symmetries of order two, i.e., involutions. LetInvF(C) ⊆Sym(C) be the set of involutions that stabilizeFpointwise. For linear 1-perfect codes, the possibilities for the number of fixed points |F| are given, establishing lower and upper bounds. For anym≥ 2 and any valuekbetween these bounds, [m/2] ≤k≤m-1, linear 1-perfect codes of lengthn=2m-1 which have an involution that fixes |F| = 2k-1 coordinates are constructed. Moreover, for anym≥ 4, 1 ≤r≤m-1, and [m/2] ≤k≤m-1, nonlinear 1-perfect codes of lengthn=2m-1 having rankn-m+rand an involution that fixes 2k-1 coordinates are also constructed, except one case, whenm≥ 6 is even,r=m-1 andk= [m/2].

A complete classification of the perfect binary one-error-correcting codes of length 15, as well as their extensions of length 16, was recently carried out in [P. R. J. O¿stergård and O. Pottonen, ¿The perfect binary one-error-correcting codes of length 15: Part I-Classification,¿IEEE Trans. Inf. Theoryvol. 55, pp. 4657-4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via switching, as it turns out that all but two full-rank codes can be obtained through a series of such transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.

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.

For two binary codes$C_1,C_2$, define$i(C_1,C_2)=\vert C_1 \cap C_2\vert$to be their intersection number. This correspondence establishes that there exist Hadamard codes of length$2^t$, for all$t\geq 3$, with intersection number$i$if and only if$i \in \{0,2,4,\ldots,2^{t+1}-12,2^{t+1}-8, 2^{t+1}\}$. Also it is proved that for all$t\geq 4$, if there exists a Hadamard matrix of order$4s$, then there exist Hadamard codes of length$2^{t+2}s$with intersection number$i$if and only if$i \in \{0,2,4,\ldots,2^{t+3}s-12,2^{t+3}s-8,2^{t+3}s \}$.

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.

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.

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.

We investigate nonsystematic perfect binary codes of length n, establishing that nonsystematic perfect codes of length n exist for all admissible $n \geq 15$. This improves on the result of Avgustinovich and Solov'eva [ Proc. 5th Internat. Workshop Algebraic and Combinatorial Coding Theory, Cosopol, Bulgaria, 1996, pp. 15--19] who have established the existence of such codes for $n\geq 255$. We also provide the results of a computer investigation of nonsystematic perfect codes of length 15.

The authors present a construction of binary self-dual codes from Eulerian graphs and establish that the code will be indecomposable if and only if the vertices of degree 2 are not a cutset of the graph. The construction is used to establish that every finite group is isomorphic to the automorphism group of some self dual code. It is further shown that deciding isomorphism of self-dual codes is at least as difficult as graph isomorphism.

Various linear and nonlinearR(r,m)codes having parameters(2^{m}, 2^{k}, 2^{m-r})withk=\sum_{i=0}^{r}\left(^{m}_{i}\right)are constructed fromR(r,q)andR(r,p)codes,m=p+q. A dual construction forR(m-r,m)codes fromR(p-r,p)andR(q-r,q)codes is also presented,m=p+q. As a simple corollary we have that the number of nonequivalentR(r,m)codes is at least exponential in the length (forr>1). ForR(m-r,m)codes, the lower bound is doubly exponential in the length (forr>1).

A general product construction for perfect single-error-correcting codes over an arbitrary alphabet is presented. Given perfect single-error-correcting codes of lengthsn, m, andq + 1over an alphabet of orderq, one can construct perfect single-error-correcting codes of length(q - 1)nm + n + mover the same alphabet. Moreover, if there exists a perfect single-error-correcting code of lengthq + 1over an alphabet of orderq, then there exist perfect single-error-correcting codes of lengthn,n = (q^{t} _ 1)/(q - 1), and(t > 0, an integer). Finally, connections between projective planes of orderqand perfect codes of lengthq + 1over an alphabet of orderqare discussed.