Weight Hierarchies Research Articles

Weight hierarchies of linear codes satisfying the almost chain condition

The weight hierarchy of a linear [n, k; q] codeC overGF(q) is the sequence (d 1,d 2, …,d k) whered r is the size of the smallest support of anr-dimensional subcode ofC. An [n, k; q] code satisfies the chain condition if there exists subcodesD 1⊂D 2⊂…⊂D k=C ofC such thatD r has dimensionr and support of sized r for allr. Further,C satisfies the almost chain condition if it does not satisfy the chain condition, but there exist subcodesD r of dimensionr and support of sized r for allr such thatD 2⊂D 3⊂…⊂D k=C andD 1⊂D 3. A simple necessary condition for a sequence to be the weight hierarchy of a code satisfying the almost chain condition is given. Further, explicit constructions of such codes are given, showing that in almost all cases, the necessary conditions are also sufficient.

A new kind of geometric structures determining the chain good weight hierarchies

In this paper, by using a new kind of geometric structures, we present some sufficient conditions to determine the weight hierarchies of linear codes satisfying the chain condition.

On the dual of quaternary Goethals codes and their weight hierarchies

Some properties such as type, trace description, symmetrized weight enumerator, and generalized Hamming weights of the dual of quaternary Goethals codes are discussed. Furthermore, the shortened codes of quaternary Goethals codes and their dual codes are proved to be cyclic.

Upper bounds on weight hierarchies of extremal non-chain codes

The weight hierarchy of a linear [ n, k; q] code C over GF(q) is the sequence ( d 1, d 2,…, d k ) where d r is the smallest support weight of an r-dimensional subcode of C . Linear codes may be classified according to a set of chain and non-chain conditions, the extreme cases being codes satisfying the chain condition (due to Wei and Yang) and extremal, non-chain codes (due to Chen and Kl øve). This paper gives upper bounds on the weight hierarchies of the latter class of codes.

Weight hierarchies of binary linear codes of dimension 4

The weight hierarchy of a binary linear [ n, k] code C is the sequence ( d 1, d 2,…, d k ) where d r is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes. The possible weight hierarchies in each class are given. For one class the details of the proofs are included.

On Weight Hierarchies of Product Codes The Wei-Yang Conjecture and more

Abstract The weight of a code is the number of coordinate positions where no codeword is zero. The r th minimum weight d r is the least weight of an r -dimensional subcode. Wei and Yang conjectured a formula for the minimum weights of some product codes. The conjecture is proved in two different ways, each with interesting side-results.

Weight hierarchies of linear codes of dimension 3

Abstract The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d1,d2,…,dk), where dr is the smallest support of an r-dimensional subcode of C. The weight hierarchies of [n,3;q] codes are studied. In particular, for q⩽5 the possible weight hierarchies of [n,3;q] codes are determined.

The puzzle of Kashmiri stress: implications for weight theory

There are currently many proposals for analysing cross-linguistic vowel length and consonant weight patterns within an Optimality Theory (OT) framework. The basic claim is that vowel length and consonant weight are determined by the interaction of markedness constraints on moraic content and constraints requiring faithfulness to underlying moraicity.In this paper, I will show how the constraints used in one such proposal (Morén 1999) provide an analysis of the core syllable weight of Kashmiri, and that the inclusion of several other constraints proposed in the literature allows us to account for the previously puzzling distribution of stress in this language. Besides the empirical importance of both providing an analysis of Kashmiri stress and weight and demonstrating a language type predicted by Morén's weight typology, an added theoretical point is the demonstration that closed syllables may vary in weight depending on surface stress assignment. This is in line with work of Kager (1989), Hayes (1995), Rice (1996) and Broselow et al. (1997), but in contrast with many other theories which treat consonant weight for a particular segment in a given syllabic position as static within a given language. Further, the OT analysis presented here provides the necessary tools to allow the expression of weight hierarchies, while maintaining the restrictive theory of weight offered by a single-tiered moraic approach.

The weight hierarchies of outer product codes

The weight hierarchy of the outer product of two self-complementary codes is expressed as a function of the weight hierarchies of the component codes. An alternative method to calculate the rth generalized Hamming weight (and so the weight hierarchy) of the first-order Reed–Muller code is given. The outer product of a self-complementary code and a nonself-complementary code is also discussed.

Competition for Milk in the Domestic Rabbit: Survivors Benefit from Littermate Deaths

We sought evidence for postnatal resource limitation among littermates of the domestic rabbit Oryctolagus cuniculus, and asked whether deaths of indi~vidual pups benefit survivors by increasing their share of milk. Milk ingestion, growth and mortality of 10 chinchilla‐breed captive litters were recorded between birth and age 21 d. That milk limits growth and survival was indicated by larger litters showing lower weight gain and higher mortality, and by a significant positive correlation between milk ingested by individual pups and weight increase. Within litters, pups with higher birth weights grew faster, and weight hierarchies became increasingly stable over the 3 weeks, suggesting that advantages accrued during gestation were progressively consolidated during lactation. After individual pups died, the total daily milk weight obtained by the litter was generally unaffected but per capita milk consumption and growth of surviving pups increased, and increases in per capita consumption were greater in smaller litters. The most successful competitors for milk apparently benefit from the deaths of their littermates by obtaining an increased share of an undiminished daily food supply. This relationship has not previously been demonstrated in any vertebrate.

The weight hierarchies of some product codes

We determine the weight hierarchies of the product of an n-tuple space and an arbitrary code, the product of an m-dimensional even-weight code and the [24,12,8] extended Golay code, and the product of an m-dimensional even-weight code and the [8,4,4] extended Hamming code. The conjecture d/sub r/=d*/sub r/ is proven for all three cases.

The Weight Hierarchy of Hermitian Codes

We compute the complete weight hierarchies of all Hermitian codes. The tools used are the arithmetic of Hermitian curves and the order bound on the generalized Hamming weights.

Generalized Hamming weights of nonlinear codes and the relation to the Z/sub 4/-linear representation

We give a new definition of generalized Hamming weights of nonlinear codes and a new interpretation connected with it. These generalized weights are determined by the entropy/length profile of the code. We show that this definition characterizes the performance of nonlinear codes on the wire-tap channel of type II. The new definition is invariant under translates of the code, it satisfies the property of strict monotonicity and the generalized Singleton bound. We check the relations between the generalized weight hierarchies of Z/sub 4/-linear codes and their binary image under the Gray map. We also show that the binary image of a Z/sub 4/-linear code is a symmetric, not necessarily rectangular code. Moreover, if this binary image is a linear code then it admits a twisted squaring construction.

Weight hierarchies of extremal non-chain binary codes of dimension 4

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d/sub 1/,d/sub 2/,/spl middot//spl middot//spl middot/,d/sub k/) where d/sub r/ is the smallest support of an r-dimensional subcode of C. An [n,k;q] code is extremal nonchain if, for any r and s, where 1/spl les/r<s/spl les/k, there are no subspaces D and E such that D/spl sub/E, dim D=r, dim E=s,w/sub S/(D)=d/sub r/, and w/sub S/(E)=d/sub s/. The possible weight hierarchies of such binary codes of dimension 4 are determined.

Immunocompetence and Nestling Survival in the House Martin: The Tasty Chick Hypothesis

In altricial birds post-fledging survival is usually positively related to nestling body mass. A large number of studies have shown that the latest hatched chick is the more likely to die, even if food is abundant. Here we suggest that ectoparasites may be a key factor in the evolution and the maintenance of the establishment of weight hierarchies within broods. We propose the hypothesis that weight hierarchies within broods may be adaptive if the chick in poor condition is the one with the least efficient immune system within a nest. In this case parasites would preferentially feed on such a tasty chick, because it would allow high reproductive rates for the parasites, without negatively affecting the survival of the other nestlings. This could prevent entire nest failure of the brood or allow the other chicks to grow more efficiently. This hypothesis was investigated in a colony of house martins Delichon urbica. We predicted that immunocompetence was positively correlated with body condition, and that nestlings dying before fledging should have lower immune responses when challenged with an antigen. T-cell immune response to an experimentally injected antigen was strongly positively related to body condition. Non-surviving chicks had low body condition and a weak immune response. The implications of these results are discussed in the context of the adaptive significance of hatching asynchrony.

Growth rate of ostrich (Struthio camelus) chicks under intensive management in Botswana.

Thirty-seven ostrich chicks raised artificially had their weights, lengths of metatarsus and heights measured weekly from hatching to the 16th week. Despite weight loss during the first week due to utilization of egg yolk by the chicks, the chicks showed and exponential growth up to the twelfth week with an overall mean weekly weight gain of 1.3 kg. Female chicks showed a superior mean weekly gain of 1.6 kg compared to 1.2 kg for males. The metatarsal length grew rapidly at a weekly rate of 2.5 cm, but began to decline in the 11th week even though the weights and heights of the chicks were still increasing. The correlation coefficient between body weight and metatarsal length was 0.97. Individual chicks also maintained their respective heavy or light weight hierarchies throughout the study period. Thus isolation of chicks into groups by weight and raising them separately could have a role in the artificial rearing of ostrich chicks. Furthermore, ostrich chick rearing may be improved by regularly weighing chicks as a means of detecting changes in the growth patterns.

Weight Hierarchies of Linear Codes Satisfying the ChainCondition

The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d_1,d_2, m,d_k) where d_r is the smallest support of an r–dimensional subcode of C. By explicit construction, it is shown that if a sequence (a_1,a_2, m,a_k) satisfies certain conditions, then it is the weight hierarchy of a code satisfying the chain condition.

The weight hierarchies and generalized weight spectra of the projective codes from degenerate quadrics

The weight hierarchies and generalized weight spectra of the projective codes from degenerate quadrics in projective spaces over finite fields are determined. These codes satisfy also the chain conditions.

Bounds on the Weight Hierarchies of Extremal Non-chain Codes of Dimension 4

The weight hierarchy of a linear [n, k; q] code C over GF(q) is the sequence (d 1, d 2, . . . , d k ) where d r is the smallest support of an r-dimensional subcode of C. An [n, k; q] code is external non-chain if for any r and s, where 1≦r<s≦k, there are no subspaces D and E, such that D⊂E, dim D=r, dim E=s, w S (D)=d r , and w S (E)=d s . Bounds on the weight hierarchies of such codes of dimension 4 are studied.

The weight hierarchies of some product codes

Bounds on the weight hierarchies of the product of two simplex codes, two first order Reed-Muller codes, and the product of a simplex code and a first-order Reed-Muller code are determined. The weight hierarchies of the product of two Hamming codes and the product of a Hamming code and an even-weight code are also discussed.