On Perfect Codes and Tilings: Problems and Solutions

A construction of perfect binary codes is presented. It is shown that this construction gives rise to perfect codes that are nonequivalent to any of the previously known perfect codes. Furthermore, perfect codes C 1 and C 2 are constructed such that their intersection C 1∩C 2 has the maximum possible cardinality. The latter result is then employed to explicitly construct 22cn nonequivalent perfect codes of length n, for sufficiently large n and some constant c slightly less than 0.5.KeywordsParity CheckParity Check MatrixCoset RepresentativePerfect CodePerfect SegmentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Perfect binary codes: bounds and properties

We prove that arbitrary n-ary quasigroups of order 4 can be transformed into each other by successive switchings of {a, b}-components. We prove that perfect (closely packed) binary codes with distance 3 whose rank (dimension of the linear span) is greater by 1 or 2 than the rank of a linear perfect code can be taken to each other by successive switchings of i-components.

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.

It is well known that a matched filter gives the maximum possible output signal-to-noise ratio (SNR) when the input is a scattering signal from a point like radar target in the presence of white noise. However, a matched filter produces unwanted sidelobes that can mask vital information. Several researchers have presented various methods of dealing with this problem. They have employed different kinds of less optimal filters in terms of the output SNR from a point-like target than that of the matched filter. In this paper we present a method of designing codes, called perfect and almost perfect pulse compression codes, that do not create unwanted sidelobes when they are convolved with the corresponding matched filter. We present a method of deriving these types of codes from any binary phase radar codes that do not contain zeros in the frequency domain. Also, we introduce a heuristic algorithm that can be used to design almost perfect codes, which are more suitable for practical implementation in a radar system. The method is demonstrated by deriving some perfect and almost perfect pulse compression codes from some binary codes. A rigorous method of comparing the performance of almost perfect codes (truncated) with that of perfect codes is presented.

Perfect nonbinary AN codes with distance three

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.

The structure of perfect binary single-error-correcting codes of length n=2/sup t/-1 is investigated. The concepts of local and interweight spectra of a code are introduced. They are generalizations of the notion of the weight spectrum of a code. Properties of the spectra of perfect codes are studied. The concept of strong distance-invariance of a code is introduced and it is shown that a perfect binary code is strong distance-invariant.

We construct a class of perfect ternary constant-weight codes of length 2^r, weight 2^r-1 and minimum distance 3. The codes have 2^{2^r-1} codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes.

We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.

J. Astola (ibid., vol. IT-32, p.443-5, May 1986) constructed new binary perfect arithmetic codes considering the modular distance introduced by T.R.N. Rao and O.N. Garcia (1971). The commenter states that, if the modular distance defined by W.E. Clark and J.J. Liang (1973) is considered, these codes are no longer perfect.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Binary CRCs are very effective for error detection, but their software implementation is not very efficient. Thus, many binary nonCRC codes (which are not as strong as CRCs, but can be more efficiently implemented in software) are proposed as alternatives to CRCs. The nonCRC codes include WSC, CXOR, one's-complement checksum, Fletcher checksum, and block-parity code. We present a general algorithm for constructing a family of binary error-detection codes. This family is large because it contains all these nonCRC codes, CRCs, perfect codes, as well as other linear and nonlinear codes. In addition to unifying these apparently disparate codes, our algorithm also generates some nonCRC codes that have minimum distance 4 (like CRCs) and efficient software implementation.

The first examples of perfect $e$-error correcting $q$-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$-ary codes. Thecase of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem. &nbsp;&nbsp; This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

A contribution is made to the classification of lattice-like total perfect codes in integer latticesn via pairs (G,�) formed by abelian groups G and homomorphisms � : Z n → G. A conjecture is posed that the cited contribution covers all possible cases. A related conjecture on the unfinished work on open problems on lattice-like perfect dominating sets inn with induced components that are parallel paths of length > 1 is posed as well.

The class of binary quadratic residue (QR) codes of length n=8m-1 contains two perfect codes. These are the (7,4,3) Hamming code and the (23,12,7) Golay code. However, it is proved in the present paper that there are no quasi-perfect QR codes of length 8m-1. Finally, this result is generalized to all binary self-dual codes of length N>72. >