• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.821-827
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• 2012

In this paper, we prove that for all odd prime powers $q$ there exist MDS(maximum distance separable) self-dual codes over $\mathbb{F}_{q^2}$ for all even lengths up to $q+1$. Additionally, we prove that there exist MDS self-dual codes of length four over $\mathbb{F}_q$ for all $q$ > 2, $q{\neq}5$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.971-997
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• 2018

The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime p are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^n$, $n{\geq}1$ over the quaternary field $F_4$ are obtained. These codes are observed to be subcodes of Q codes of length $p^n$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a Q code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic Q codes of order 6 exist.

• Journal of Communications and Networks
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• v.5 no.2
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• pp.174-183
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• 2003

Space-Time (ST) codes are known to provide high transmission rates, diversity and coding gains. In this paper, a tight upper bound on the error probability of ST codes over rapid fading channels is presented. Moreover, ST codes suitable for rapid fading channels are presented. These codes are designed using the QPSK and 16-QAM signal constellations. The proposed codes are based on two different encoding schemes. The first scheme uses a single trellis encoder, whereas the second scheme uses the I-Q encoding technique. Code design is achieved via partitioning the signal space such that the design criteria are maximized. As a solution for the decoding problem of I-Q ST codes, the paper introduces a low-complexity decoding algorithm. Results show that the I-Q ST codes using the proposed decoding algorithm outperform singleencoder ST codes with equal complexity. The proposed codes are tested over fading channels with different interleaving conditions, where it is shown that the new codes are robust under such imperfect interleaving conditions.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1237-1252
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• 2023

A fundamental problem in coding theory is to find n_q(k, d), the minimum length n for which an [n, k, d]_q code exists. We show that some q-divisible optimal linear codes of dimension 4 over 𝔽_q, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG(3, q). We also construct some new linear codes over 𝔽_q with q = 7, 8, which determine n₇(4, d) for 31 values of d and n₈(4, d) for 40 values of d.

• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.609-619
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• 2019

The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order q are presented by La Garcia in [14]. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in [14].

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.915-923
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• 2015

Self-dual codes have been actively studied because of their connections with other mathematical areas including t-designs, invariant theory, group theory, lattices, and modular forms. We presented the building-up construction for self-dual codes over GF(q) with $q{\equiv}1$ (mod 4), and over other certain rings (see [19], [20]). Since then, the existence of the building-up construction for the open case over GF(q) with $q=p^r{\equiv}3$ (mod 4) with an odd prime p satisfying $p{\equiv}3$ (mod 4) with r odd has not been solved. In this paper, we answer it positively by presenting the building-up construction explicitly. As examples, we present new optimal self-dual [16, 8, 7] codes over GF(7) and new self-dual codes over GF(7) with the best known parameters [24, 12, 9].

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.965-989
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• 2015

Let p, ${\ell}$ be distinct odd primes, q be an odd prime power with gcd(q, p) = gcd(q,${\ell}$) = 1, and m, n be positive integers. In this paper, we determine all self-dual, self-orthogonal and complementary-dual negacyclic codes of length $2p^{n{\ell}m}$ over the finite field ${\mathbb{F}}_q$ with q elements. We also illustrate our results with some examples.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ 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. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ 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 case n = 2,3 is dealt separately.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1513-1521
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• 2013

In this paper we will study cyclic codes over some special rings: $\mathbb{F}_q[u]/(u^i)$, $\mathbb{F}_q[u_1,{\ldots},u_i]/(u^2_1,u^2_2,{\ldots},u^2_i,u_1u_2-u_2u_1,{\ldots},u_ku_j-u_ju_k,{\ldots})$, and $\mathbb{F}_q[u,v]/(u^i,v^j,uv-vu)$, where $\mathbb{F}_q$ is a field with $q$ elements $q=p^r$ for some prime number $p$ and $r{\in}\mathbb{N}-\{0\}$.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.385-397
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• 2009

We study codes over the polynomial ring ${\mathbb{F}}_q[D]$ and their projections to the finite rings ${\mathbb{F}}_q[D]/(D^m)$ and the weight enumerators of self-dual codes over these rings. We also give the formula for the number of codewords of minimum weight in the projections.