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We present three constructions of partial difference sets (PDS) using different types of finite local rings. The first construction uses homomorphic images of finite Frobenius local rings and generalizes a previous result by the author. The second construction uses finite Frobenius local rings. The third construction uses finite (noncommutative) chain rings and generalizes a recent construction of partial difference sets by Leung and Ma. The three constructions provide many new PDS in nonelementary abelian groups.

Using a lifting formula for the coefficients of Boolean functions, we characterize binary resilient functions as binary matrices with certain row or column intersection properties. We give some new constructions of binary resilient functions based on this characterization. In particular, we show that the incidence matrix of a Steiner system can be used to construct binary resilient functions.

A theorem of Kneser states that in an abelian group G, if A and B are finite subsets in G and AB={ab:a∈A,b∈B}, then ∣AB∣⩾∣A∣+ ∣B∣− ∣H(AB)∣ where H(AB)={g∈G:g(AB)=AB}. Motivated by the study of a problem in finite fields, we prove an analogous result for vector spaces over a field E in an extension field K of E. Our proof is algebraic and it gives an immediate proof of Kneser's Theorem.

Latin square type partial difference sets (PDS) are known to exist in R × R for various abelian p-groups R and in ℤt. We construct a family of Latin square type PDS in ℤt × ℤ2ntp using finite commutative chain rings. When t is odd, the ambient group of the PDS is not covered by any previous construction. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 394–402, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10029

Let $q$ be a prime power. We determine all permutation trinomials of $\Bbb F_{q^2}$ of the form $ax+bx^q+x^{2q-1}\in\Bbb F_{q^2}[x]$. The subclass of such permutation trinomials of $\Bbb F_{q^2}$ with $a,b\in\Bbb F_q$ was determined in a recent paper by the author., Comment: 16 pages

A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. A finite chain ring, roughly speaking, is an extension over a Galois ring of characteristic pnusing an Eisenstein polynomial of degree k. When p∤k, such rings were classified up to isomorphism by Clark and Liang. However, relatively little is known about finite chain rings when p∣k. In this paper, we allowed p∣k. When n=2 or when p∥k but (p−1)∤k, we classified all pure finite chain rings up to isomorphism. Under the assumption that (p−1)∤k, we also determined the structures of groups of units of all finite chain rings.

We generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the Teichmüller sets of the Galois rings. Let R=GR(p2, t) be the Galois ring of characteristic p2 and rank t with Teichmüller set T and let π:R→R/pR be the natural homomorphism. We give a construction of PDS in R with the parameters ν=p2t, k=r(pt−1), λ=pt+r2−3r, μ=r2−r, where r=lpt−s(p, t), 1≤l≤ps(p, t), and s(p, t) is the largest dimension of a GF(p)-subspace W⊂R/pR such that π−1(W)∩T generates a subgroup of R of rank

Building sets are a successful tool for constructing semi-regular divisible difference sets and, in particular, semi-regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi-regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000

We give two generalizations of some known constructions of relative difference sets. The first one is a generalization of a construction of RDS by Chen, Ray-Chaudhuri and Xiang using the Galois ring i>GR(4, i>m). The second one generalizes a construction of RDS by Ma and Schmidt from the setting of chain rings to a setting of more general rings.

Bent functions and partial difference sets have been constructed from finite principal ideal local rings. In this paper, the constructions are generalized to finite quasi-Frobenius local rings. Let R be a finite quasi-Frobenius local ring with maximal ideal M. Bent functions and certain partial difference sets on M\times M are extended to R\times R.

We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on Z 2 m {\mathbf Z}_2^m .

A chain ring is a finite local principal ideal ring.q-ary bent functions are constructed from chain rings which are freeZq-modules. A construction combining such bent functions is also given.

In this paper, we present three results on bent functions: a construction, a restriction, and a characterization. Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables. The restriction imposes new conditions on the directional derivatives of bent functions. Certain non-existence results that were previously obtained through computer search follow easily from these conditions. The characterization describes bent functions as certain solutions of a system of quadratic equations. Interesting new properties of bent functions are obtained using the characterization.

Let M n ( F ) be the algebra of all n × n matrices over an arbitrary field F , and for S , T ⊂ M n ( F ), let C T ( S ) = B ∈ T : AB = BA for all A ∈ S . For each A ∈ M n ( F ), it is well known that C M n ( F ) ( A )) = F [ A ], the algebra of polynomials in A over F . We determine C M n ( F ) ( C GL( n , F ) ( A )). It turns out that C M n ( F ) ( C GL( n , F ) ( A )) = F [ A ] when | F | > 2. When vbF vb = 2, C M n ( F ) ( C GL( n , F ) ( A )) = F [ A ] unless A has both 0 and 1 as eigenvalues and the elementary divisors of greatest degree corresponding to 0 and 1 are not repeated in the list of elementary divisors. In this exceptional case, C M n ( F ) ( C GL( n , F ) ( A )) = F [ A ] ⊕ R , where R is an explicitly described two dimensional subspace of M n ( F ). We also determine C GL( n , F ) ( C GL( n , F ) ( A )).

We prove that the Reed-Muller code R(1,7) is normal. The normality of R(1,7) was a long-standing open question, and has an important consequence in a conjecture about the function t[n,k], the smallest covering radius of any [n,k] binary linear code.

Let $R$ be a commutative chain ring. We use a variation of Gr\"obner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]/I$ Frobenius? When is $R[x]/I$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $\text{rad}\,R$ is small, the algorithms provide explicit answers to the questions., Comment: 24 pages, 3 figures

We study various spaces of magic squares over a field, and determine their dimensions. These results generalize the main result of Small from 1988.

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and $\text{Tr}_{q/2}(\frac 1t)=0$; (ii) $q\equiv 1\pmod 8$ and $t^2=-2$., 18 pages

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained., Comment: 4 pages

A deep result about the Reed–Muller codes, proved by Mykkeltveit in 1980, is that the covering radius of the Reed–Muller codeR(1, 7) equals 56. We discover an alternative and simpler proof for this important result.

We prove that the covering radius of the Reed-Muller code R(1,9) in R(4,9) is 240, not exceeding the quadratic bound.

Let AGL(m, F) be the general affine group of order m over an arbitrary field F. We determine the conjugacy classes of AGL(m, F). When F = GF(q), we also determine the sizes of the centralizers of the elements in AGL(m, F). The group AGL(m, 2) (= AGL(m, GF(2))) acts on each of the Reed-Muller codes of length 2m as an automorphism group. We denote the rth order Reed-Muller code of length 2m by R(r, m) and prove that under the action of AGL(m, 2) the number of orbits of R(t, m)/R(s, m) is equal to that of R(m − (s + 1), m)/R(m − (t + 1), m) for −1 ≤ s < t ≤ m. We also compute the numbers of AGL(m, 2)-orbits of R(t, m)/R(s, m) for m = 6, 7, and −1 ≤ s < t ≤ m.

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation polynomial (PP) of $\Bbb F_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\em desirable} if $g_{n,q}$ is a PP of $\Bbb F_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP., 47 pages, 3 tables

Let $q>2$ be a prime power and $f={\tt x}^{q-2}+t{\tt x}^{q^2-q-1}$, where $t\in\Bbb F_q^*$. It was recently conjectured that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following holds: (i) $t=1$, $q\equiv 1\pmod 4$; (ii) $t=-3$, $q\equiv \pm1\pmod{12}$; (iii) $t=3$, $q\equiv -1\pmod 6$. We confirm this conjecture in the present paper., 10 pages

Cosets of the Reed-Muller code R ( m −3, m ) are classified under the actions of GL( m ,2) and GA( m ,2), the latter being the automorphism group of R ( m —3, m ) for m ⩾4. The number of cosets in each class is calculated. Orphans of R ( m −3, m ) are identified, and the normality of R ( m −3, m ) is established. A recursive formula is given for computing weight distributions of cosets of R ( m −3, m ). The formula gives the number of vectors of weight w in a coset of R ( m −3, m ) with minimal weight μ rather easily when w is not far away from μ.

A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A, c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms of pointed abelian groups, and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.

In this paper we report the following computational results on partial spread functions in eight variables: (i) the numbers of equivalence classes of partial spread functions (in eight variables) of all possible orders; (ii) the total number of partial spread bent functions in eight variables; (iii) the distribution of the cardinalities of stabilizers (in GL(8, F2)) of partial spread bent functions in eight variables. The computational method is also described.

LetR(r, m) by therth order Reed-Muller code of length2 m , and let ρ(r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. ρ(r+1,m+2)≥ 2ρ(r, m)+2 if 0≤r≤m−2. This improves the successive use of the known inequalities ρ(r+1,m+2)≥2ρ(r+1,m+1) and ρ(r+1,m+1)≥ρ (r, m).2.ρ(2, 7)≤44. Previously best known upper bound for ρ(2, 7) was 46. 3. The covering radius ofR(1,m) inR(m−1,m) is the same as the covering radius ofR(1,m) inR(m−2,m) form≥4.

Let R(r,m) be the rth-order Reed-Muller code of length 2/sup m/ and let rho (r,m) be its covering radius. R

Let R(r, m) be the rth order Reed-Muller code of length 2m, and let ρ(r, m) be its covering radius. We prove that if 2≤ k ≤ m - r - 1, then ρ(r + k, m + k) ≥ ρ(r, m + 2(k - 1). We also prove that if m - r ≥ 4, 2 ≤ k ≤ m - r - 1, and R(r, m) has a coset with minimal weight ρ(r, m) which does not contain any vector of weight ρ(r, m) + 2, then ρ(r + k, m + k) ≥ ρ(r, m) + 2k(. These inequalities improve repeated use of the known result ρ(r + 1, m + 1) ≥ ρ(r, m).

LetG be a finite abelian group,K a subfield ofC, C[G] regarded as an algebra of matrices.A G K {A?C[G]| all the entries and eigenvalues ofA are inK} is an association algebra overK. In this paper, the association scheme ofA G K is determined and in the caseK=Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved thatZ 4×Z 4×Z 4 is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.

Let $Q_{2^m}$ be the generalized quaternion group of order $2^m$ and $D_N$ the dihedral group of order $2N$. We classify the orbits in $Q_{2^m}^n$ and $D_{p^m}^n$ ($p$ prime) under the Hurwitz action.

Some new lower bounds on mod C mod for a binary linear (n, k)R code C with n+1=t(R+1)-r(0 2 odd) or with n+1=t(R+1)-1(t>2 even) are obtained. These bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t(n, k), the smallest covering radius of any (n, k) code. >

Let $\Psi_{k,n}$ denote the number of inequivalent binary self-orthogonal $[n,k]$ codes. We present a method which allows us to compute $\Psi_{k,n}$ explicitly for a moderate $k$ and an arbitrary $n$. Included in this paper are explicit formulas for $\Psi_{k,n}$ with $k\le 5$., Comment: 41 pages

Let $\Omega$ be an algebraic closure of ${\mathbb Q}_p$ and let $F$ be a finite extension of ${\mathbb Q}_p$ contained in $\Omega$. Given positive integers $f$ and $e$, the number of extensions $K/F$ contained in $\Omega$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number ${\mathfrak I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine ${\mathfrak I}(F,f,e)$ completely when $p^2\nmid e$ and get partial results when $p^2\parallel e$. When $s$ is large, ${\mathfrak I}({\mathbb Q}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\mathbb F}_{p^f}$, ramification index $e$, and length $s$., Comment: 39 pages

Two upper bounds for the norm N(C) of a binary linear code C with minimal weight d and covering radius R are given. The second of these bounds implies that C is normal if R=3. >

Let C be a binary code (not necessarily linear) with covering radius R and length n=3R+2. The sphere covering bound on the cardinality of C is improved considerably provided C has minimal distance d>2. Some new results on the function t(n,k) (the smallest covering radius of any binary linear code with length n and dimension k): t(38.6)>or=13, t(47.7)>or=16, t(59.8)>or=20 are given. >

Let Λ = ℤ[t, t-1] be the ring of Laurent polynomials over ℤ. We classify all Λ-modules M with |M| = pn, where p is a prime and n ≤ 4. Consequently, we have a classification of Alexander quandles of order pn for n ≤ 4.

We consider the sum of the reciprocals of all monic polynomials of a given degree over a finite field 픽q each raised to the power of k. When k ≤ q, the sum has a surprisingly simple result due to mysterious cancellations that occur in the sum. We discuss this interesting phenomenon and provide a new inductive proof.

Let p be a prime and let r, e and m be positive integers such that r|e and e|m. The enumeration of linear codes of length p m over double struck capital F p r which are invariant under the affine linear group AGL(m/e, double struck capital F p e)is equivalent to the enumeration of certain ideals in a partially ordered set (U, e and p 3)-invariant linear codes of length p m over double struck capital F p r.

After a discussion of automorphisms of Reed-Muller codes the authors show that the Reed-Muller code R(r,m) is not Z/sub 4/-linear for 3/spl les/r/spl les/m-2, proving a conjecture by Hammons, Kumar, Calderbank, Sloane, and Sole (1994).

Whether quasi-perfect codes are normal is addressed. Let C be a code of length n, dimension k, covering radius R, and minimal distance d. It is proved that C is normal if d>or=2R-1. Hence all quasi-perfect codes are normal. Consequently, any (n,k)R binary linear code with minimal distance d>or=2R-1 is normal. >

Let R(r,m) be the r th order Reed-Muller code of length 2 m . For −1⩽r⩽s⩽m , the action of the general affine group AGL (m,2) on R(s,m)/R(r,m) defines a linear association scheme on R(s,m)/R(r,m) . In this paper, we determine the eigenmatrix of the linear association scheme on R(2,m) (=R(2,m)/R(−1,m) ). Our approach relies on the Mobius inversion and detailed calculations with the general linear group and the symplectic group over GF (2) . As a consequence, we obtain explicit formulas for the weight enumerators of all cosets of R(m−3,m) . Such explicit formulas were not available previously.