• ETRI Journal
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• 제28권4호
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• pp.453-460
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• 2006

In this article, we introduce a new class of product codes based on convolutional codes, called convolutional product codes. The structure of product codes enables parallel decoding, which can significantly increase decoder speed in practice. The use of convolutional codes in a product code setting makes it possible to use the vast knowledge base for convolutional codes as well as their flexibility in fast parallel decoders. Just as in turbo codes, interleaving turns out to be critical for the performance of convolutional product codes. The practical decoding advantages over serially-concatenated convolutional codes are emphasized.

• ETRI Journal
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• 제32권4호
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• pp.588-595
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• 2010

This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N-dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well-known nonlinear Nordstrom-Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of $10^{-6}$ when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom-Robinson codes has an effective gain of about 0.7 dB at a BER of $10^{-5}$ when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi-cyclic codes.

• Journal of applied mathematics & informatics
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• 제32권3_4호
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• pp.529-537
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• 2014

The paper introduces a new product of polynomials defined over a field. It is a generalization of the ordinary product with inner polynomial getting non-overlapping segments obtained by multiplying with coefficients and variable with expanding powers. It has been called 'Ordered Power Product' (OPP). Considering two rings of polynomials $R_m[x]=F[x]modulox^m-1$ and $R_n[x]=F[x]modulox^n-1$, over a field F, the paper then considers the newly introduced product of the two polynomial rings. Properties and algebraic structure of the product of two rings of polynomials are studied and it is shown to be a ring. Using the new type of product of polynomials, we define a new product of two cyclic codes and devise a method of getting a cyclic code from the 'ordered power product' of two cyclic codes. Conditions for the OPP of the generators polynomials of component codes, giving a cyclic code are examined. It is shown that OPP cyclic code so obtained is more efficient than the one that can be obtained by Kronecker type of product of the same component codes.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제10권7호
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• pp.3353-3370
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• 2016

The way of combination of Product Perfect Codes (PPCs) is based on the theory of short codes constructing long codes. PPCs have larger embedding rate than Hamming codes by expending embedding columns in a coding block, and they have been proven to enhance the performance of the F₅ steganographic method. In this paper, the proposed modified product codes called MPCs are introduced as an efficient way to embed more data than PPCs by increasing 2^r₂-1-r₂ embedding columns. Unlike PPC, the generation of the check matrix H in MPC is random, and it is different from PPC. In addition a simple solving way of the linear algebraic equations is applied to figure out the problem of expending embedding columns or compensating cases. Furthermore, the MPCs over ℤ₄ have been proposed to further enhance not only the performance but also the computation speed which reaches O(n₁+σ). Finally, the proposed ℤ₄-MPC intends to maximize the embedding rate with maintaining less distortion , and the performance surpasses the existing improved product perfect codes. The performance of large embedding rate should have the significance in the high-capacity of covert communication.

• 한국통신학회논문지
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• 제37권7A호
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• pp.512-521
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• 2012

본 논문에서는 곱부호를 위한 연판정 데이터의 결합을 이용한 복호 기법을 유한 비트 연판정만을 적용하여 사용할 수 있도록 새로운 연판정 데이터의 결합 법칙을 제시한다. 기존의 무한 연판정을 이용한 복호 기법은 복잡한 tanh 연산에 기반을 두고 있으므로 높은 연산 복잡도와 함께 하드웨어에 적용하기 어려운 단점이 있다. 따라서 기존 복호 기법에서 이용되는 연산들을 분석하여 작성한 간단한 연산 테이블을 이용하여 복호하는 방법을 제시한다. 또한 연판정 데이터를 적용하기 용이한 길쌈부호를 곱부호의 수평부호로 이용하였을 때 연판정 데이터의 결합 방식을 찾는 방법을 제시하고 모의 실험을 통해 성능을 검증한다. 제안된 알고리듬은 4비트 유한 연판정을 적용한 길쌈부호를 곱부호의 수평부호로 이용하였을 때 무한 연판정을 적용한 경우의 성능에 근접함을 보였다.

• ETRI Journal
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• 제31권5호
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• pp.518-524
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• 2009

Single parity check (SPC) product codes are simple yet powerful codes that are used to correct errors and/or recover erasures. The focus of this paper is to evaluate the performance of such codes under erasure scenarios and to develop a closed-form tight upper bound for the post-decoding erasure rate. Closed-form exact expressions are derived for up to seven erasures. Previously published closed-form bounds assumed that all unrecoverable patterns should contain four erasures in a square. Additional non-square patterns are accounted for in the proposed expressions. The derived expressions are verified using exhaustive search. Eight or more erasures are accounted for by using a bound. The developed expressions improve the evaluation of the recoverability of SPC product codes without the need for simulation or search algorithms, whether exhaustive or novel.

• 한국통신학회논문지
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• 제32권3C호
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• pp.205-215
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• 2007

This paper presents low latency and/or computation algorithms of iterative codes of turbo codes, turbo product codes and low density parity check codes for use in wireless broadband communication systems. Due to high coding complexity of iterative codes, this paper focus on lower complexity and/or latency algorithms that are easily implementable in hardware and further accelerate the decoding speed.

• 한국통신학회논문지
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• 제26권4B호
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• pp.427-436
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• 2001

일반적인 통신 시스템에서 시스템의 성능을 저하시키는 오류는 검출 불능 오류와 정정 불능 오류로 구성되는데 검출 불능 오류의 경우 오류에 대한 적절한 대응이 불가능하다는 측면에서 더욱 심각한 문제가 된다. 이러한 검출 불능 오류를 줄이기 위해 적부호 복호에는 다양한 방법이 생각되었지만 이러한 방법들의 대부분은 CD, 혹은 DAT등에 적용하기 위한 방법들로서 긴 길이의 RS 적부호에 적용하기 힘들다. 본 논문에서는 검출 불능 오류율을 줄일 수 있는 긴 길이의 RS 적부호 복호 방법을 제안하고 이들의 성능을 심볼 오류 확률을 계산함으로서 분석하여 기존의 방법과 비교한다. 여기서 제안된 방법들은 기존의 방법보다 검출 불능 확률을 크게 줄이는데 반하여 전체 오류 확률에서의 큰 열화는 일으키지 않는다.

• 대한수학회보
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• 제55권2호
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• pp.341-357
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• 2018

We examine various dualities over the fields of even orders, giving new dualities for additive codes. We relate the MacWilliams relations and the duals of ${\mathbb{F}}_{2^{2s}}$ codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order $2^s$ of the additive group is a self-dual code with respect to some duality.

• 한국통신학회논문지
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• 제29권2C호
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• pp.181-186
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• 2004

최근 터보 부호에 비해서 구현시 복잡하지 않고, 높은 부호화율에서 거의 샤논 이론에 접근하는 Turbo Product Code에 대해 관심이 고조되고 있다. 본 논문에서는 기존의 Turbo Product code 복호기에서 row과 column을 직렬로 복호를 하지 않고 복호 구조가 병렬로 동작하는 Turbo Product code 복호기를 제안한다. 모의 실험을 한 결과 기존의 방식에 비해 복호 지연이 줄어들고 성능면에서 직렬 방식과 거의 비슷한 성능이 나타난다.