Binary Linear Codes With Optimal Scaling: Polar Codes With Large Kernels

Binary Linear Codes with Optimal Scaling and Quasi-Linear Complexity

We present the first family of binary codes that attains optimal scaling and quasi-linear complexity, at least for the binary erasure channel (BEC). In other words, for any fixed δ > 0, we provide codes that ensure reliable communication at rates within ε > 0 of the Shannon capacity with block length n = O(1/ε2+δ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n). Furth...

Optimal binary linear locally repairable codes with disjoint repair groups

In recent years, several classes of codes are introduced to provide some fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) play an important role. However, most known constructions are over large fields with sizes close to the code length, which lead to the systems computationally expensive. Due to this, binary...

Binary Optimal Linear Rate 1/2 Codes

Optimal Linear Codes Over GF(7) and GF(11) with Dimension 3

Let $n_q(k,d)$ denote the smallest value of $n$ for which there exists a linear $[n,k,d]$-code over the Galois field $GF(q)$. An $[n,k,d]$-code whose length is equal to $n_q(k,d)$ is called {em optimal}. In this paper we present some matrix generators for the family of optimal $[n,3,d]$ codes over $GF(7)$ and $GF(11)$. Most of our given codes in $GF(7)$ are non-isomorphic with the codes pre...

Soft-decision decoding of polar codes with Reed-Solomon kernels

The problem of efficient soft-decision decoding of polar codes with ReedSolomon kernel is considered. A decomposition of the kernel based on the cyclotomic FFT algorithm is proposed, which enables one to implement near-optimal evaluation of log-likelihood ratios in the successive cancellation decoding algorithm.