Turbo codes with symmetric and asymmetric component codes defined over finite fields of integers

Binary asymmetric turbo codes and non-binary turbo codes have been proposed to improve the bit error rate (BER) performance of parallel concatenated coding schemes. Both strategies have certain advantages that can be exploited when they are put together. This paper investigates turbo codes based on two component recursive systematic convolutional (RSC) codes defined over a finite field of integers. Symmetric and asymmetric non-binary turbo codes are obtained and their BER performance in both the `waterfall? and the `error-floor? regions is analysed. The results show good performance improvements when compared to binary and quaternary turbo codes with same throughput.     

Dual codes of systematic group codes over abelian groups

For systematic codes over finite fields the following result is well known: If [I¦P] is the generator matrix then the generator matrix of its dual code is [ ?P ^tr¦I]. The main result is a generalization of this for systematic group codes over finite abelian groups. It is shown that given the endomorphisms which characterize a group code over an abelian group, the endomorphisms which characterize its dual code are identified easily. The self-dual codes are also characterized. It is shown that there are self-dual and MDS group codes over elementary abelian groups which can not be obtained as linear codes over finite fields.     

Dual codes of systematic group codes over abelian groups

For systematic codes over finite fields the following result is well known: If [I∣P] is the generator matrix then the generator matrix of its dual code is The main result is a generalization of this for systematic group codes over finite abelian groups. It is shown that given the endomorphisms which characterize a group code over an abelian group, the endomorphisms which characterize its dual code are identified easily. The self-dual codes are also characterized. It is shown that there are self-dual and MDS group codes over elementary abelian groups which can not be obtained as linear codes over finite fields. Received March 7, 1995; revised version April 3, 1996     

On quasi-cyclic codes over {\mathbb{Z}_q}

Quasi-cyclic (QC) codes are a remarkable generalization of cyclic codes. Many QC codes have been shown to be best for their parameters. In this paper, some structural properties of QC codes over the prime power integer residue ring ${\mathbb{Z}_q}$ are considered. An l-QC code of length lm over ${\mathbb{Z}_q}$ is viewed both as in the conventional row circulant form and also as a ${\frac{\mathbb{Z}_q[x]}{\langle x^m-1 \rangle}}$ -submodule of ${\frac{GR(q,l)[x]}{\langle x^m-1 \rangle}}$ , where GR(q, l) is the Galois extension ring of degree l over ${\mathbb{Z}_q}$ . A necessary and sufficient condition for cyclic codes over Galois rings to be free is obtained and a BCH type bound for them is also given. A sufficient condition for 1-generator QC codes to be ${\mathbb{Z}_q}$ -free is given and a formula to evaluate their ranks is derived. Some distance bounds for 1-generator QC codes are also discussed. The duals of QC codes over ${\mathbb{Z}_q}$ are also briefly discussed.     

Constructions of self-dual codes and formally self-dual codes over rings

We shall describe several families of X-rings and construct self-dual and formally self-dual codes over these rings. We then use a Gray map to construct binary formally self-dual codes from these codes. In several cases, we produce binary formally self-dual codes with larger minimum distances than known self-dual codes. We also produce non-linear codes which are better than the best known linear codes.     

Matrix-product codes over finite chain rings

Tim Blackmore and Graham H. Norton introduced the notion of matrix-product codes over finite fields. The present paper provides a generalization to finite chain rings. For codes a distance function is defined using a homogeneous weight function in the ring. It is proved that the minimum distance of a matrix-product codes is determined by the minimum distances of the separate codes. At the end of the paper we focus on Galois rings and define a special family of matrix-product codes.      

Higher weights for codes over rings

A generalized definition of higher weights for codes over finite chain rings and principal ideal rings and bounds on the minimum higher weights in this setting are given. Using this we generalize the definition for higher MDS and MDR codes. Computationally, the higher weight enumerator of lifted Hamming and Simplex codes over mathbbZ₄{mathbb{Z}_4}, the minimum higher weights for the lifted code of the binary [8,4,4] self-dual extended Hamming code, the lifted code of the ternary [12,6,6] self-dual Golay code and the lifted code of the binary [24,12,8] self-dual Golay code are given. Joint weight enumerators are used to produce MacWilliams relations for specific higher weight enumerators.     

Scroll codes over curves of higher genus

We construct linear codes from scrolls over curves of high genus and study the higher support weights d _i of these codes. We embed the scroll into projective space ${\mathbb{P}^{k-1}}$ and calculate bounds for the d _i by considering the maximal number of ${\mathbb{F}_q}$ -rational points that are contained in a codimension h subspace of ${\mathbb{P}^{k-1}}$ . We find lower bounds of the d _i and for the cases of large i calculate the exact values of the d _i . This work follows the natural generalisation of Goppa codes to higher-dimensional varieties as studied by S.H. Hansen, C. Lomont and T. Nakashima.     

Euclidean self-dual codes over non-commutative Frobenius rings

We study self-dual codes over non-commutative Frobenius rings. It is shown that a code is equal to its left orthogonal if and only if it is equal to its right orthogonal. Constructions of self-dual codes are given over Frobenius rings that arise from self-dual codes over the center of the ring. These constructions are used to show for which lengths self-dual codes exist over various rings.     

On cyclic self-dual codes

We investigate cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We give a decomposition of a repeated-root cyclic codes over \mathbbF_p^r{\mathbb{F}_{p^{r}}} . The decomposition is used to analyze cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} . We obtain a necessary and sufficient condition for the existence of nontrivial cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} , and prove that all cyclic self-dual codes over \mathbbF_2^r{\mathbb{F}_{2^{r}}} are Type I. Finally we classify cyclic self-dual codes of some lengths over \mathbbF₄{\mathbb{F}_{4}} , \mathbbF₈{\mathbb{F}_{8}} , and \mathbbF₁₆{\mathbb{F}_{16}} .     

On cyclic codes which areq-ary images of linear codes

This paper presents a complete characterization of cyclic codes over GF(q) which areq-ary images of linear codes over GF(q ²). New cyclic codes over GF(2 ^r ) are constructed as images of other cyclic codes over GF(2^3r ), for some positive integersr. An application to decoding is given.     

1-generator quasi-cyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring with $1\ne 0$ . 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let $\gamma $ be a fixed generator of the maximal ideal of R, $F=R/\langle \gamma \rangle $ and $|F|=q$ . For any positive integers m, n satisfying $\mathrm{gcd}(q,n)=1$ , let $\mathcal{R}_n=R[x]/\langle x^n-1\rangle $ . Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator $\mathcal{R}_n$ -submodules of the module $\mathcal{R}_n^m$ . First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that $\mathrm{gcd}(|q|_n,m)=1$ , where $|q|_n$ denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.     

On the equivalence of linear codes

Certain equivalent conditions are given for the equivalence of linear codes using the properties of the relative subcodes derived from the relative generalized Hamming weight.     

On the dimension of Goppa codes

A lower bound for the dimension of geometric BCH codes (i.e. subfield subcodes of Goppa codes) has been given by M. Wirtz [7]. We prove that this bound is actually exact for small enough divisorG.     

Rotational invariance of two-level group codes over dihedral and dicyclic groups

Phase-rotational invariance properties for two-level constructed, (using a binary code and a code over a residue class integer ring as component codes) Euclidean space codes (signal sets) in two and four dimensions are discussed. The label codes are group codes over dihedral and dicyclic groups respectively. A set of necessary and sufficient conditions on the component codes is obtained for the resulting signal sets to be rotationally invariant to several phase angles.     

Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration

For R a Galois ring and m ₁, . . . , m _l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} is an R[x]-submodule of R[x]/(x^m₁-1)×?×R[x]/(x^m_l-1){R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}. Suppose m ₁, . . . , m _l are all coprime to the characteristic of R and let {g ₁, . . . , g _t } be the set of all monic basic irreducible polynomials in the factorizations of x^m_i-1{x^{m_i}-1} (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m ₁, m ₂, . . . , m _l ) and length ?_i=1^lm_i{\sum_{i=1}^lm_i} are identified with G₁×?×G_t{{\mathcal G}_1\times\cdots\times {\mathcal G}_t}, where G_j{{\mathcal G}_j} is an R[x]/(g _j )-submodule of (R[x]/(g_j))^n_j{(R[x]/(g_j))^{n_j}}, where n _j is the number of i for which g _j appears in the factorization of x^m_i-1{x^{m_i}-1} into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.     

Adaptive RS codes for message delivery over an encrypted mobile network

The authors present a hybrid automatic repeat request technique using adaptive Reed-Solomon (RS) codes with packet erasure. This technique suits the transport layer in tactical mobile wireless networks with type I encryption, where encryption erasures the entire Internet protocol packet. The novelty of the presented technique is the multifaceted optimisation of Reed-Solomon codes at the transport layer for delivery assurance, speed of service (SoS) and network throughput. With this technique, the transport layer in tactical networks can meet the stringent requirements of quality of service imposed by the tactical network user, even under adverse conditions. These requirements define a high level of reliability (delivery assurance), a specific SoS and optimum use of the limited bandwidth (BW) of the wireless network, where the probability of packet erasure can be very high. The provided probabilistic analysis shows that focusing on network throughput alone will result in violating SoS and delivery assurance requirements. On the other hand, focusing on SoS and delivery assurance requirements can result in poor network throughput. The multifaceted optimisation technique, which utilises hybrid ARQ for message delivery, is described using a homogeneous Markov chain.