Lloyd's Theorem Research Articles

The non-existence of some perfect codes over non-prime power alphabets

Let exp p ( q ) denote the number of times the prime number p appears in the prime factorization of the integer q . The following result is proved: If there is a perfect 1-error correcting code of length n over an alphabet with q symbols then, for every prime number p , exp p ( 1 + n ( q − 1 ) ) ≤ exp p ( q ) ( 1 + ( n − 1 ) / q ) . This condition is stronger than both the packing condition and the necessary condition given by the Lloyd theorem, as it for example excludes the existence of a perfect code with the parameters ( n , q , e ) = ( 19 , 6 , 1 ) .

Tight 2-designs and perfect 1-codes in Doob graphs

Abstract Doob graphs are distance-regular graphs having the same parameters as the quaternary Hamming graphs. Delsarte's generalization of Lloyd's theorem implies that a tight 2e -design or a perfect e -code in a Doob graph can possibly exist only when e=1 . We construct perfect 1 -codes in Doob graphs of diameter 5 , and tight 2 -designs in all Doob graphs of diameter (4 l −1)/3 .

Tilings of Binary Spaces

We study partitions of the space $\mathbb{F}_2^n $ of all the binary n-tuples into disjoint sets, where each set is an additive cosec of a given set V. Such a partition is called a tiling of $\mathbb{F}_2^n $ and denoted $(V,A)$, where A is the set of cosec representatives. We give a sufficient condition for a set V to be a tile in terms of the cardinality of $V + V$. We then employ this condition to classify all tilings with sets of small cardinality. Further, periodicity of tilings in $\mathbb{F}_2^n $ is discussed, and a simple construction of nonperiodic tilings of $\mathbb{F}_2^n $ is presented for all $n \geq 6$. It is also shown that the nonperiodic tiling of $\mathbb{F}_2^6 $ is unique. A tiling $(V,A)$ is said to be proper if V generates $\mathbb{F}_2^n $; it is said to be full rank if both V and A generate $\mathbb{F}_2^n $. We show that, in general, the classification of tilings can be reduced to the study of proper tilings. We then prove that any tiling may be decomposed into smaller tilings that are either trivial or have full rank. Existence of full-rank tilings is exhibited by showing that each tiling is uniquely associated with a perfect binary code. Moreover, it is shown that periodic full-rank tilings may be further decomposed into smaller tilings, and then the existence of nonperiodic full-rank tilings is deduced. Finally, we generalize the well-known Lloyd theorem, originally stated for tilings by spheres, for the case of arbitrary tilings.

Completely regular codes

Completely regular codes are a large class of codes which share many of the most interesting properties of perfect codes. This paper shows that the basic theory—including Lloyd's theorem—can be obtained very elegantly by methods inspired from the theory of distance regular graphs.

A Lloyd Theorem in Weakly Metric Association Schemes

Weakly metric association schemes are a generalization of metric schemes occurring in graph theory (vertex transitive digraphs), coding theory (Lee scheme, Clark-Liang scheme), and group theory (conjugacy scheme). Linear constraints on the inner distribution of r-covering codes in these schemes lead to a ‘Lloyd theorem’ on perfect codes. This generalizes results from Delsarte, Biggs, Bassalygo and Landauer.

Weight distribution of translates, covering radius, and perfect codes correcting errors of given weights

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> bed binary linear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,k)</tex> code defined by a check matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> and let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(x)</tex> be the characteristic function for the set of columns of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> . Connections between the Walsh transform of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(x)</tex> and the weight distributions of all translates of the code are obtained. Explicit formulas for the weight distributions of translates are given for small weights <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i(i&lt;8)</tex> . The computation of the weight distribution of all translates (including the code itself) for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i&lt;8</tex> requires at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7(n-k)2^{n-k}</tex> additions and subtractions, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6 \cdot 2^{n-k}</tex> multiplications and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n-k+l}</tex> memory cells. This method may be very effective if there is an analytic expression for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(x)</tex> . A simple method for computing the covering radius of the code by the Walsh transform of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(x)</tex> is described. The implementation of this method requires for large <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n-k}(n-k) \log_{2}(n-k)</tex> arithmetical operations and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n-k+1}</tex> memory cells. We define the concept <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -perfect for codes, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> is a set of weights. After describing several linear and nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -perfect codes, necessary and sufficient conditions for a code to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -perfect in terms of the Walsh transform of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h(x)</tex> are established. An analog of the Lloyd theorem for such codes is proved.

Perfect codes in the graphs Ok and L(Ok)

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.

Perfect codes in the graphs Ok and L(Ok)

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of e because of the difficulty of working with the sphere packing condition. For e= 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e=4 and we shall prove that there are no perfect 4-codes in Ok.

Lloyd's theorem for perfect codes

We give another proof of Lloyd's theorem using homogeneous distance enumerators, and show that the same techniques will give similar theorems for more general types of codes. The theorem has been proved earlier by Delsarte and Lenstra. We thought it interesting that the result can be derived from some elementary polynomial manipulations. The methods and results herein should be considered as belonging to combinatorial coding theory, since it is not necessary to use the finite field approach to get them.

An elementary proof of Lloyd's theorem

An elementary proof of Lloyd's theorem

An improved version of Lloyd's theorem

The generalisation of Lloyd's theorem to distance-transitive graphs can be improved in the case of antipodal graphs by looking at the derived graph. In the case of binary perfect codes the roots of the Lloyd polynomial are even integers. This can be applied to give a short proof of the binary perfect code theorem.

A generalized Lloyd theorem and mixed perfect codes.

A generalized Lloyd theorem and mixed perfect codes.

Perfect codes in graphs

The classical problem of the existence of perfect codes is set in a vector space. In this paper it is shown that the natural setting for the problem is the class of distance-transitive graphs. A general theory is developed that leads to a simple criterion for the existence of a perfect code in a distance-transitive graph, and it is shown that this criterion implies Lloyd's theorem in the classical case.

On the Nonexistence of Perfect Codes over Finite Fields

It is proved that there are no unknown perfect (Hamming-)error-correcting codes over finite fields.

Two theorems on perfect codes

Two theorems are proved on perfect codes. The first one states that Lloyd's theorem is true without the assumption that the number of symbols in the alphabet is a prime power. The second theorem asserts the impossibility of perfect group codes over non-prime-power- alphabets.