Perfect binary codes: bounds and properties

Although nontrivial perfect binary codes exist only for length n = 2m -1 with $m \ge 3$ and for length n=23, many interesting problems concerning these codes remain unsolved. Herein, we present solutions to some of these problems. In particular, we show that the smallest nonempty intersection of two perfect codes of length 2m -1 consists of two codewords, for all $m \ge 3$. We also provide a complete solution to the intersection number problem for Hamming codes. Furthermore, we prove that for $m \ge 3$, a perfect code of length 2m-1 -1 is embedded in a perfect code $\Bbb{C}$ of length 2m -1 if and only if ${\Bbb C}$ is not of full rank. This result implies the existence of distinct generalized Hamming weights for perfect codes, and we determine completely the generalized Hamming weights of all perfect codes that do not contain embedded full-rank perfect codes. We further explore the close ties between perfect codes and tilings: we prove that full-rank tilings of ${\Bbb F}_{2}^n$ exist for all $n \geq 14$ and show that the existence of full-rank tilings for other n is closely related to the existence of full-rank perfect codes with kernels of high dimension. We briefly survey the present state of knowledge on perfect binary codes and list several interesting and important open problems concerning perfect codes and tilings.

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.

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.

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.

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.

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 present a few new constructions for perfect linear single byte-correcting codes. These constructions generate some perfect single byte-correcting codes with new parameters, and some perfect single byte-correcting codes with known parameters and simpler presentation and implementation over the known codes. It is also shown that nonequivalent perfect linear single byte-correcting codes exist when all the bytes have the same size.

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.    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.

Preface. On Discrete Approximations of Continuous Functions with Bounded Second Derivative G.G. Amanzhaev. On a Property of Perfect Binary Codes S.V. Avgustinovich. Edge Neighborhoods in Normal Maps S.V. Avgustinovich, O.V. Borodin. An Upper Bound for the Length of a Snake in the n-Dimensional Unit Cube P.G. Emel'yanov. Operations and Isometric Embeddings of Graphs Related to the Metric Prolongation Property T.I. Fedoryaeva. Effective Algorithms for Solving Multi-Level Plant Location Problem E.Kh. Gimadi. On the Complexity of Realization of Boolean Functions by Circuits over an Infinite Basis O.M. Kasim-Zade. Dynamic Problems of Choosing Optimal Composition of a System of Technical Tools Yu.A. Kochetov, M.G. Pashchenko. Lower Bounds in the Problem of Choosing an Optimal Composition of a Two-Level System of Technical Tools Yu.A. Kochetov, M.G. Pashchenko. On Schedules of a Single Machine Jobs with Processing Times Nonlinear in Time A.V. Kononov. Problems of Reconstructing Words from Their Fragments and Applications V.K. Leont'ev. A Vertex Variant of the Kleitman-West Problem B. List. Regular Graphs Whose Every Edge Belongs to Many Triangles A.A. Makhnev. On Closed Classes in k-Valued Logic Which Contain a Homogeneous Switching Function S.S. Marchenkov. Strongly Essential Variables of the Functions of k-Valued Logic I. Mirchev. On Comparison Between the Sizes of Read-k-Times Branching Programs E.A. Okol'nishnikova. Some Optimization Problems of Scheduling the Transmission of Messages in a Local Communication Network A.V. Pyatkin. The Maximum-Weight Traveling Salesman Problem in Finite-Dimensional Real Spaces A.I. Serdyukov. Nonstrict Vector Summation in the Plane and Its Applications to Scheduling Problems S.V. Sevast'yanov.Bounds for Approximate Solution of a Scheduling Problem P.I. Sharygin. On Certain Bounds for the Weight of l- Balanced Boolean Functions Yu.V. Tarannikov. Spectral Properties of Perfect Binary (n,3)-Codes A.Yu. Vasile'va. Distributive Coloring of Graph Vertices V.G. Vizing. Deciphering Threshold Functions of k-Valued Logic N.Y. Zolotykh, V.N. Shevchenko. Index.

We construct at least \(\frac{1}{{8n^2 \sqrt 3 }}e^{\pi \sqrt {2n/3} } (1 + o(1))\) pairwise nonequivalent transitive extended perfect codes of length 4n as n → ∞.

The intersection problem for Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -additive codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , the abelian group structure of the intersection codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> cap C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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.

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.