On the admissible families of components of hamming codes

In this paper, the properties of the i-components of Hamming codes are described. We suggest constructions of the admissible families of components of Hamming codes. Each q-ary code of length m and minimum distance 5 (for q = 3, the minimum distance is 3) is shown to embed in a q-ary 1-perfect code of length n = (q ^m − 1)/(q − 1). Moreover, each binary code of length m+k and minimum distance 3k + 3 embeds in a binary 1-perfect code of length n = 2 ^m − 1.     

On nonsystematic perfect codes over finite fields

The nonsystematic perfect q-ary codes over finite field F _q of length n = (q ^m − 1)/(q − 1) are constructed in the case when m ≥ 4 and q ≥ 2 and also when m = 3 and q is not prime. For q ≠ 3, 5, these codes can be constructed by switching seven disjoint components of the Hamming code H _q ⁿ ; and, for q = 3, 5, eight disjoint components.     

A larger GL2 large sieve in the level aspect

In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL ₂ cusp forms modular with respect to the congruence subgroups Γ₁(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q ^2−δ for any δ > 0, where N is the length of linear forms in the large sieve.     

Optimal and perfect difference systems of sets from <Emphasis Type="Italic">q</Emphasis>-ary sequences with difference-balanced property

Difference systems of sets (DSS) are important for the construction of codes for synchronization. In this paper, a general construction of optimal and perfect difference systems of sets based on q-ary sequences of period n = −1 (mod q) with difference- balanced property is presented, where q is a prime power. This works for all the known q-ary sequences with ideal autocorrelation, and generalizes the earlier construction based on ternary sequences with ideal autocorrelation. In addition, we construct another class of optimal and perfect difference systems of sets, employing decimation of q-ary d-form sequences of period q ^m −1 with difference-balanced property, which generalizes the previous construction from power functions.     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

A lower bound for the computation complexity of a <Emphasis Type="Italic">q</Emphasis>-ary counter of multiplicity <Emphasis Type="Italic">Q</Emphasis> in the class of <Emphasis Type="Italic">π</Emphasis>-circuits

A generalization of the concept of parallel-sequential switching circuits (π-circuits) to the case when the variables assigned to contacts can take not two, as in the Boolean case, but a greater number of values. The conductivity of the contact is still two-valued (the contact is either closed or open). A lower bound is obtained on the complexity of these circuits computing the q-ary counter of multiplicity q, i.e., the function φ _q : {0, 1, …, q − 1} ⁿ → {0, 1} that equals 1 if the sum of values of its variables is a multiple of q.     

Density of smooth maps in Wk, p(M, N) for a close to critical domain dimension

Assuming m − 1 < kp < m, we prove that the space C ^∞(M, N) of smooth mappings between compact Riemannian manifolds M, N (m = dim M) is dense in the Sobolev space W ^k,p (M, N) if and only if π _m−1(N) = {0}. If π _m−1(N) ≠ {0}, then every mapping in W ^k,p (M, N) can still be approximated by mappings M → N which are smooth except in finitely many points.     

<Emphasis Type="Italic">mth</Emphasis>-Root subgeometry partitions

New subgeometry partitions of PG(n − 1, q ^m ) by subgeometries isomorphic to PG(n − 1, q) are constructed.      

Caps with free pairs of points

We say that two points x, y of a cap C form a free pair of points if any plane containing x and y intersects C in at most three points. For given N and q, we denote by m₂⁺ (N, q) the maximum number of points in a cap of PG(N, q) that contains at least one free pair of points. It is straightforward to prove that m₂⁺ (N, q) ≤ (q^N-1 + 2q − 3)/(q − 1), and it is known that this bound is sharp for q = 2 and all N. We use geometric constructions to prove that this bound is sharp for all q when N ≤ 4. We briefly survey the motivation for constructions of caps with free pairs of points which comes from the area of statistical experimental design. Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and MITACS NCE of Canada.     

Gaussian partitions

Let V=V(n,q) denote the finite vector space of dimension n over the finite field with q elements. A subspace partition of V is a collection Π of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. In a recent paper, we proved some strong connections between the lattice of the subspace partitions of V and the lattice of the set partitions of n={1,…,n}. We now define a Gaussian partition of [n] _q =(q ⁿ −1)/(q−1) to be a nonincreasing sequence of positive integers formed by ordering all elements of some multiset {dim(W):W∈Π}, where Π is a subspace partition of V. The Gaussian partition function gp(n,q) is then the number of all Gaussian partitions of [n] _q , and is naturally analogous to the classical partition function p(n). In this paper, we initiate the study of gp(n,q) by exhibiting all Gaussian partitions for small n. In particular, we determine gp(n,q) as a polynomial in q for n≤5, and find a lower bound for gp(6,q).     

Complete classification of the positive solutions of −Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We study the equation −Δu + u ^q = 0, q > 1, in a bounded C ² domain Ω ⊂ ℝ ^N . A positive solution of the equation is moderate if it is dominated by a harmonic function and σ-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the subcritical case, 1 < q <, q _c = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More recently, Dynkin proved, by probabilistic methods, that this remains valid in the supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this paper, we prove that for all q ≥ q _c , every positive solution is σ-moderate. We use purely analytic techniques, which apply to the full supercritical range. The main tools come from linear and non-linear potential theory. Combined with previous results, our result establishes a one-to-one correspondence between positive solutions and their boundary traces in the sense of [36].     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

The sum of digits of Gaussian primes

Let q=−a±i and denote by s _q the complex sum-of-digits function. We show that the sequence (αs _q (p)) running over all Gaussian primes lying in a circular sector is uniformly distributed modulo 1 if and only if α is irrational. Moreover, we prove that the sum-of-digits function of primes is well distributed in arithmetic progressions. This work generalizes a theorem of Mauduit and Rivat that was the solution of a long-standing conjecture by Gelfond concerning the usual q-ary sum-of-digits function. It improves also a result of Drmota, Rivat, and Stoll, who could only deal with sufficiently large prime bases q=−a±i and the full disc.     

Finiteness properties of certain arithmetic groups in the function field case

It is proved that the finiteness length of Γ=SL _n (ℱ _q [t]) isn−2 ifn≥2 andq≥2 ⁿ⁻². The proof consists in studying the homotopy type of a certain Γ-invariant filtration of an appropriate Bruhat-Tits building on which Γ acts.     

On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

 A t(v,k,λ) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly λ blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(q ⁿ +q ⁿ⁻¹ +⋯+q+1,q ²+q+1, q ⁿ⁻² + q ⁿ⁻³ +⋯+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,λ) design is said to be resolvable if the blocks can be partitioned as ℱ={R ₁,R ₂,…,R _s }, where s=λ(v−1)/(k−1) and each R _i consists of v/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length v on the points and ℱ^σ=ℱ, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=〈σ〉 where σ is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group. Received: November 10, 1999 Final version received: September 18, 2000 Acknowledgments. The author would like to thank A. Munemasa for his assistance in writing computer programs on constructing projective spaces and searching for partial spreads. Moreover, she's thankful to T. Hishida and M.␣Jimbo for helpful discussions and for verifying the results of this paper. Present address: Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City 1108, Philippines. e-mail: jumela@mathsci.math.admu.edu.ph     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

The f(q) mock theta function conjecture and partition ranks

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N_e(n) (resp. N_o(n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincaré series whose “holomorphic part” is q^-1f(q²⁴), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N_e(n) and N_o(n). Mathematics Subject Classification (2000) 11P82, 05A17     

On Intersection Problem for Perfect Binary Codes

The main result is that to any even integer q in the interval 0 ≤ q ≤ 2^{n+1-2log(n+1)}, there are two perfect codes C₁ and C₂ of length n = 2^m − 1, m ≥ 4, such that |C₁ ∩ C₂| = q.     

A lower bound for the r -order of a matrix modulo N

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of \Bbb F_qⁿ-{\Bbb F}_{q^n}- rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).