A new extension theorem for linear codes

For an [n,k,d]_q code with k3, gcd(d,q)=1, the diversity of is defined as the pair (Φ₀,Φ₁) with

All the diversities for [n,k,d]_q codes with k3, d−2 (mod q) such that A_i=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated.     

On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

Permutations with Low Discrepancy Consecutive k-sums

Consider the permutation π=(π₁,…, π_n) of 1,2,…, n as being placed on a circle with indices taken modulo n. For given kn there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k+6, independent of n. For g= gcd(n,k)>1, we show that the discrepancy is . For g=1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n.We also give an analysis for the easier case of linear permutations, where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n,k.     

Classifying Subspaces of Hamming Spaces

A linear code in F ⁿ _q with dimension k and minimum distance at least d is called an [n, k, d] _q code. We here consider the problem of classifying all [n, k, d] _q codes given n, k, d, and q. In other words, given the Hamming space F ⁿ _q and a dimension k, we classify all k-dimensional subspaces of the Hamming space with minimum distance at least d. Our classification is an iterative procedure where equivalent codes are identified by mapping the code equivalence problem into the graph isomorphism problem, which is solved using the program nauty. For d = 3, the classification is explicitly carried out for binary codes of length n 14, ternary codes of length n 11, and quaternary codes of length n 10.     

On subspace arrangements of type

Let denote the subspace arrangement formed by all linear subspaces in given by equations of the form

₁x_i1=₂x_i2==_kx_ik,

where 1i₁<<i_kn and (₁,…,_k){+1,−1}^k.Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In a previous work on a larger class of subspace arrangements by Björner and Sagan (J. Algebraic Combin. 5 (1996) 291–314) the topology of the intersection lattice turned out to be a particularly interesting and difficult case.We prove in this paper that Pure(Π_n,k^±) is shellable, hence that Π_n,k^± is shellable for k>n/2. Moreover, we prove that unless i≡n−2 (mod k−2) or i≡n−3 (mod k−2), and that is free abelian for i≡n−2 (mod k−2). In the special case of Π_2k,k^± we determine homology completely. Our tools are generalized lexicographic shellability, as introduced in Kozlov (Ann. Combin. 1 (1997) 67–90), and a spectral sequence method for the computation of poset homology first used in Hanlon (Trans. Amer. Math. Soc. 325 (1991) 1–37).We state implications of our results on the cohomology of the complements of the considered arrangements.     

Reflection sieve obstructions in finite groups of type dn

In [4] we constructed certain homology representations of a finite group G of type A_n, B_n or C_n, and showed that these representations can be used to sift out the reflection compound characters of G. In the present note, we show that for a group G of type D_n, each reflection compound character π^(k), 2 k n − 2, determines a unique “obstruction” character θ^(k), which occurs with positive multiplicity in every homology representation containing π^(k).     

Discrete Approximation of Unbounded Operators and Approximation of their Spectra

Let E be a Banach space over and let the densely defined closed linear operator A: (A)E→E be discretely approximated by the sequence ((A_n, (A_n)))_n of operators A_n where each A_n is densely defined in the Banach space F_n. Let σ_a(A) be the approximate point spectrum of A and let σ(A_n) denote the -pseudospectrum of A_n. Generalizing our own result, we show that σ_a(A)lim inf σ(A_n)=_n ∩_kn σ(A_k) holds for every >0. We deduce that then for every compact set K lim_n dist(σ_a(A)∩K, σ_a(A_n))=0 provided there exists M>0 such that (λ−A_n)⁻¹M dist(λ, σ(A_n))⁻¹ holds for every n and every λ in the resolvent set ρ(A_n) of A_n. We finally treat the problem under which conditions σ_a(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩_>0 ∩_{n kn} σ_, a(A_k)σ_a(A) hold where σ_, a(A) denotes the -approximate pseudospectrum?     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

A Coding Theory Bound and Zero-Sum Square Matrices

For a code C=C(n,M) the level k code of C, denoted C _k , is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive integer divisible by 8, and n=k, M=k2k then there is a codeword in C _k whose weight is either 0 or at most . In particular, if <(4–2)²/48 then there is a codeword in C _k whose weight is n/2–(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k,p) denote the minimum integer guaranteeing that in any square matrix over Z _p , of order f(k,p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f(k,2)/k<3.836. For general p we obtain, using a different approach, that f(k,p)p( ^{k / ln k )(1+ o} _k ⁽¹⁾⁾.     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

Twisting geometric codes

The aim of this paper is to explain how, starting from a Goppa code C(X,G,P₁,…,P_n) and a cyclic covering π:Y→X of degree m, one can twist the initial code to another one C(X,G+D_χ,P₁,…,P_n), where D_χ is a non-principal degree 0 divisor on X associated to a character χ of Gal(Y/X), in the hope that ℓ_X(G+D_χ)>ℓ_X(G). We give, using a MAGMA program, several examples where this occurs, and where both the initial and twisted codes have same minimum distance, so that initial codes have been improved.     

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

We show that any binary (n = 2 ^k − 3, 2 ^n−k , 3) code C ₁ is a cell of an equitable partition (perfect coloring) (C ₁, C ₂, C ₃, C ₄) of the n-cube with the quotient matrix ((0, 1, n−1, 0)(1, 0, n−1, 0)(1, 1, n−4, 2)(0, 0, n−1, 1)). Now the possibility to lengthen the code C ₁ to a 1-perfect code of length n + 2 is equivalent to the possibility to split the cell C ₄ into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C ₄. In any case, C ₁ is uniquely embedable in a twofold 1-perfect code of length n + 2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords. By one example, we briefly discuss 2 − (n, 3, 2) multidesigns with similar restrictions. We also show a connection of the problem with the problem of completing latin hypercuboids of order 4 to latin hypercubes.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

On the code generated by the incidence matrix of points and k-spaces in and its dual

In this paper, we study the p-ary linear code C_k(n,q), q=p^h, p prime, h1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For kn/2, we link codewords of C_k(n,q)C_k(n,q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k<n/2, we present counterexamples to lemmas valid for kn/2. Next, we study the dual code of C_k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F_p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension.     

Gray code enumeration of families of integer partitions

In this paper we show that the elements of certain families of integer partitions can be listed in a minimal change, or Gray code, order. In particular, we construct Gray code listings for the classes P_δ(n, k) and D(n, k) of partitions of n into parts of size at most k in which, for P_δ(n, k), the parts are congruent to one modulo δ and, for D(n, k), the parts are distinct. It is shown that the elements of these classes can be listed so that the only change between successive partitions is the increase of one part by δ (or the addition of δ ones) and the decrease of one part by δ (or the removal of δ ones), where, in the case of D(n, k), δ = 1.     

An Asymptotic Property of Schachermayer's Space under Renorming

Let X be a Banach space with closed unit ball B. Given k , X is said to be k-β, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence (x_n) B there are indices (n_i)^k_{i = 1}, respectively, (n_i)^{k + 1}_{i = 1}, such that (1/(k + 1))||x + ∑^k_{i = 1} x_ni|| ≤ 1 − δ, respectively, (1/(k + 1))||∑^{k + 1}_{i = 1} x_ni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.     

The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes

For a fixed integer m ≥ 0, and for n = 1, 2, 3, ..., let λ_{2m, n}(x) denote the Lebesgue function associated with (0, 1,..., 2m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2k−1) π/(2n)]: k=1, 2, ..., n}. We examine the Lebesgue constant Λ_{2m, n} max{λ_{2m, n}(x): −1 ≤ x ≤ 1}, and show that Λ_{2m, n} = λ_{m, n}(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of Λ_{2m, n)} as n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for Λ_{2, n}.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

On the Grassmann modules for the unitary groups

Let V be 2n-dimensional vector space over a field 𝕂 equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let 𝕂₀ ? 𝕂 be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, …, 2n}, there exist natural representations of the groups G ? U(2n, 𝕂/𝕂₀) and H = G ∩ SL(V) ? SU(2n, 𝕂/𝕂₀) on the k-th exterior power of V. With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.