Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p

In this paper, we investigate Hadamard matrices of order 2(p + 1) with an automorphism of odd prime order p. In particular, the classification of such Hadamard matrices for the cases p = 19 and 23 is given. Self‐dual codes related to such Hadamard matrices are also investigated. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 367–380, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10052     

A result on the weight distributions of binary quadratic residue codes

Truong et al. [7]proved that the weight distribution of a binary quadratic residue code C with length congruent to −1 modulo 8 can be determined by the weight distribution of a certain subcode of C containing only one-eighth of the codewords of C. In this paper, we prove that the same conclusion holds for any binary quadratic residue codes.     

Density of integers which are discriminants of cyclic fields of odd prime degree

An asymptotic formula is given for the number of integers x which are discriminants of cyclic fields of odd prime degree.Received: 17 February 2004     

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

The odd primary homology of SU(n), Sp(n) and Spin(n)-gauge groups

For an odd prime $p$, we calculate the mod-$p$ homology of $SU\left(n\right)$-gauge groups over a simply-connected, closed 4-manifold for all $n\ge 2$. Similar calculations are obtained for the structure groups $Sp\left(n\right)$ when $n\ge 1$ and $Spin\left(n\right)$ for $n\ge 3$ (except for some cases when $n$ is even and $p=3$).     

Generalizing binary quadratic residue codes to higher power residues over larger fields

In this paper, we provide a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order. We find generating polynomials for such codes, define a new notion corresponding to the binary concept of an idempotent, and use this to find a lower bound for the codeword weight of the duals of such codes. This in turn leads to a lower bound on the weight of the codewords themselves.     

A class of completek-caps inPG(3,q) forq an odd prime

The main problem on caps, posed originally by Segre in the fifties, is to determine the values of k for which there exists a complete k-cap. Very few results on this problem are known. The cardinality of the largest cap(s) and the smallest complete cap(s) are crucial. In this paper it is shown that there exist complete k-caps in PG(3, q), q an odd prime 5 or q = 9, such that k = (q² + q + 6)/3 or k = (q² + 2q + 6)/3. These complete caps are smaller than those currently known for q odd.In memoriam Giuseppe Tallini     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.