Perfect codes as isomorphic spaces

Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.     

A Full Rank Perfect Code of Length 31

A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last open case of the rank-kernel problem of Etzion and Vardy. AMS Classification: 94B25     

Construction of universal one-way hash functions: Tree hashing revisited

We present a binary tree based parallel algorithm for extending the domain of a universal one-way hash function (UOWHF). For t?2, our algorithm extends the domain from the set of all n-bit strings to the set of all ((2^t-1)(n-m)+m)-bit strings, where m is the length of the message digest. The associated increase in key length is 2m bits for t=2; m(t+1) bits for 3?t?6 and m×(t+⌊log₂(t-1)⌋) bits for t?7.     

Perfect codes from the dual point of view I

The two concepts dual code and parity check matrix for a linear perfect 1-error correcting binary code are generalized to the case of non-linear perfect codes. We show how this generalization can be used to enumerate some particular classes of perfect 1-error correcting binary codes. We also use it to give an answer to a problem of Avgustinovich: whether or not the kernel of every perfect 1-error correcting binary code is always contained in some Hamming code.     

On perfect <Emphasis Type="Italic">p</Emphasis>-ary codes of length <Emphasis Type="Italic">p</Emphasis> + 1

Let p be a prime number and assume p ≥ 5. We will use a result of L. Redéi to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p − 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.      

The Classification of Some Perfect Codes

Perfect 1-error correcting codes C in Z ₂ ⁿ , where n=2 ^m–1, are considered. Let ; denote the linear span of the words of C and let the rank of C be the dimension of the vector space . It is shown that if the rank of C is n–m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n–m+2, described by a Hamming code H and a set of MDS-codes D _h , h H, over an alphabet with four symbols. The case of rank n–m+1 is much simpler: Any such code is a Vasil'ev code.     

On the Steiner quadruple systems of small rank which are embeddable into extended perfect binary codes

The codewords of weight 4 of every extended perfect binary code that contains the all-zero vector are known to form a Steiner quadruple system. We propose a modification of the Lindner construction for the Steiner quadruple system of order N = 2 ^r which can be described by special switchings from the Hamming Steiner quadruple system. We prove that each of these Steiner quadruple systems is embedded into some extended perfect binary code constructed by the method of switching of ijkl-components from the binary extended Hamming code. We give the lower bound for the number of different Steiner quadruple systems of order N with rank at most N ? logN + 1 which are embedded into extended perfect codes of length N.     

A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks

In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to ζ(3) = 1/1³ + 1/2³ + 1/3³ +… as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log₂ logn+ω(1), where ω(1) is any function going to ∞ with n, then the minimum bounded-depth spanning tree still has weight tending to ζ(3) as n → ∞, and that if k < log₂ logn, then the weight is doubly-exponentially large in log₂ logn ? k. It is NP-hard to find the minimum bounded-depth spanning tree, but when k≤log₂ logn?ω(1), a simple greedy algorithm is asymptotically optimal, and when k ≥ log₂ logn+ω(1), an algorithm which makes small changes to the minimum (unbounded depth) spanning tree is asymptotically optimal. We prove similar results for minimum bounded-depth Steiner trees, where the tree must connect a specified set of m vertices, and may or may not include other vertices. In particular, when m=const×n, if k≥log₂ logn+ω(1), the minimum bounded-depth Steiner tree on the complete graph has asymptotically the same weight as the minimum Steiner tree, and if 1 ≤ k ≤ log₂ logn?ω(1), the weight tends to $(1 - 2^{ - k} )\sqrt {8m/n} \left[ {\sqrt {2mn} /2^k } \right]^{1/(2^k - 1)}$ in both expectation and probability. The same results hold for minimum bounded-diameter Steiner trees when the diameter bound is 2k; when the diameter bound is increased from 2k to 2k+1, the minimum Steiner tree weight is reduced by a factor of $2^{1/(2^k - 1)}$ .     

A generalization of the binary Preparata code

A classical binary Preparata code P₂(m) is a nonlinear (2^m+1,2^2(2m-1-m),6)-code, where m is odd. It has a linear representation over the ring Z₄ [Hammons et al., The Z₄-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319]. Here for any q=2^l>2 and any m such that (m,q-1)=1 a nonlinear code P_q(m) over the field F=GF(q) with parameters (q(Δ+1),q^2(Δ-m),d?3q), where Δ=(q^m-1)/(q-1), is constructed. If d=3q this set of parameters generalizes that of P₂(m). The equality d=3q is established in the following cases: (1) for a series of initial admissible values q and m such that q^m<2¹⁰⁰; (2) for m=3,4 and any admissible q, and (3) for admissible q and m such that there exists a number m₁ with m₁|m and d(P_q(m₁))=3q. We apply the approach of [Nechaev and Kuzmin, Linearly presentable codes, Proceedings of the 1996 IEEE International Symposium Information Theory and Application Victoria, BC, Canada 1996, pp. 31-34] the code P is a Reed-Solomon representation of a linear over the Galois ring R=GR(q²,4) code P dual to a linear code K with parameters near to those of generalized linear Kerdock code over R.     

An algebraic approach to approximate evaluation of a polynomial on a set of real points

The previous best algorithm for approximate evaluation of a polynomial on a real set was due to Rokhlin and required of the order ofmu+nu ³ infinite precision arithmetic operations to approximate [on a fixed bounded setX(m) ofm+1 real points] a degreen polynomial $p\left( z \right) = \sum\nolimits_{i = 0}^n {p_i x^i } $ within the error bound $2^{ - u} \sum\nolimits_{i = 0}^n {\left| {p_i } \right|} $ . We develop an approximation algorithm which exploits algebraic computational techniques and decreases Rokhlin's record estimate toO(mlog² u+nmin-u, logn}). For logu=o(logn), this result may also be favorably compared with the record boundO(m+n)log² n) on the complexity of the exact multipoint polynomial evaluation. The new algorithm can be performed in the fields (or rings) generated by the input values, which enables us to decrease the precision of the computations [by using modular (residue) arithmetic] and to simplify our computations further in the case whereu=O(logn). Our algorithm allows NC and simultaneously processor efficient parallel implementation. Because of the fundamental nature of the multipoint polynomial evaluation, our results have further applications to numerical and algebraic computational problems. In passing, we also show a substantial improvement in the Chinese remainder algorithm for integers based on incorporating Kaminski's fast residue computation.     

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n?=?2 ^m ? 3, 2 ^n-m-1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n?=?2 ^m ? 4, 2 ^n-m , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.     

An Efficient Algorithm for the Complex Roots Problem

Given a univariate polynomialf(z) of degreenwith complex coefficients, whose norms are less than 2^min magnitude, the root problem is to find all the roots off(z) up to specified precision 2^−μ. Assuming the arithmetic model for computation, we provide an algorithm which has complexityO(nlog⁵nlogB), whereb= χ + μ, and χ = max{n,m}. This improves on the previous best known algorithm of Pan for the problem, which has complexityO(n²log²nlog(m+ μ)). A remarkable property of our algorithm is that it does not require any assumptions about the root separation off, which were either explicitly, or implicitly, required by previous algorithms. Moreover it also has a work-efficient parallel implementation. We also show that both the sequential and parallel implementations of the algorithm work without modification in the Boolean model of arithmetic. In this case, it follows from root perturbation estimates that we need only specify θ = ⌈n(B+ logn+ 3)⌉ bits of the binary representations of the real and imaginary parts of each of the coefficients off. We also show that by appropriate rounding of intermediate values, we can bound the number of bits required to represent all complex numbers occurring as intermediate quantities in the computation. The result is that we can restrict the numbers we use in every basic arithmetic operation to those having real and imaginary parts with at most φ bits, where[formula]and[formula]Thus, in the Boolean model, the overall work complexity of the algorithm is only increased by a multiplicative factor ofM(φ) (whereM(ψ) =O(ψ(log ψ) log log ψ) is the bit complexity for multiplication of integers of length ψ). The key result on which the algorithm is based, is a new theorem of Coppersmith and Neff relating the geometric distribution of the zeros of a polynomial to the distribution of the zeros of its high order derivatives. We also introduce several new techniques (splitting sets and “centered” points) which hinge on it. We also observe that our root finding algorithm can be efficiently parallelized to run in parallel timeO(log⁶nlogB) usingnprocessors.     

Optimal binary covering codes of length 2j

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j, it is known that K(n,1) = 2 · K(n?1,1) = 2^{n ? j}. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n ? 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.     

A note on optimal multiway split trees

Split trees are suitable data structures for storing records with different access frequencies. Under assumption that the access frequencies are all distinct, Huang has proposed anO(n ⁴ logm) time algorithm to construct an (m+1)-way split tree for a set ofn keys. In this paper, we generalize Huang's algorithm to deal with the case of non-distinct access frequencies. The technique used in the generalized algorithm is a generalization of Hesteret al.'s, where the binary case was considered. The generalized algorithm runs inO(n ⁵ logm) time.     

Enumerating and decoding perfect linear Lee codes

Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Z _q , and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm.      

On the classification of perfect codes: side class structures

The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C′ are linearly equivalent if there exists a non-singular matrix A such that AC = C′ where C and C′ are matrices with the code words of C and C′ as columns. Hessler proved that the perfect codes C and C′ are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

Hadamard full propelinear codes of type Q; rank and kernel

Hadamard full propelinear codes (\(\mathrm{HFP}\)-codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of \(\mathrm{HFP}\)-codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an \(\mathrm{HFP}\)-code of length 4n, dimension of the kernel \(k=2\), and maximum rank \(r=2n\), we obtain an \(\mathrm{HFP}\)-code of double length 8n, dimension of the kernel \(k=2\), and maximum rank \(r=4n\).