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Register a new userEmbedding linear codes into self-orthogonal codes and their optimal minimum distances
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Added by
Young-Hun Kim
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d ge d(mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n otequiv 6,13,14,21,22,28,29 pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n equiv 0,3,4,5,10,11,12 pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code.
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The parameters of MDS self-dual codes are completely determined by the code length. In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual (self-orthogonal) codes and MDS almost self-dual codes over. The main idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual (self-orthogonal). The evaluation sets are consists of two subsets which satisfy some certain conditions and the length of these codes can be expressed as a linear combination of two factors of q-1. Four families of MDS self-dual codes, two families of MDS self-orthogonal codes and two families of MDS almost self-dual codes are obtained and they have new parameters.
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We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within $varepsilon > 0$ of capacity, the code length $n$ often scales as $O(1/varepsilon^{mu})$, where the constant $mu$ is called the scaling exponent. It is known that the optimal scaling exponent is $mu=2$, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the $2times 2$ kernel) on the BEC is $mu=3.63$. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist $elltimesell$ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent $mu(ell)$ that tends to the optimal value of $2$ as $ell$ grows. We furthermore characterize precisely how large $ell$ needs to be as a function of the gap between $mu(ell)$ and $2$. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity $O(n)$ and encoding/decoding complexity $O(nlog n)$.
This paper considers the problem of simultaneously communicating two messages, a high-security message and a low-security message, to a legitimate receiver, referred to as the security embedding problem. An information-theoretic formulation of the problem is presented. A coding scheme that combines rate splitting, superposition coding, nested binning and channel prefixing is considered and is shown to achieve the secrecy capacity region of the channel in several scenarios. Specifying these results to both scalar and independent parallel Gaussian channels (under an average individual per-subchannel power constraint), it is shown that the high-security message can be embedded into the low-security message at full rate (as if the low-security message does not exist) without incurring any loss on the overall rate of communication (as if both messages are low-security messages). Extensions to the wiretap channel II setting of Ozarow and Wyner are also considered, where it is shown that perfect security embedding can be achieved by an encoder that uses a two-level coset code.
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