WebThe cyclic linear code generated by this polynomial will contain all binary words that are multiples of this polynomial, where multiplication is done modulo 2. Let C be the smallest … WebNov 15, 2013 · A (binary) formally self-dual code is a linear code whose weight enumerator is equal to that of its dual. Little is known about the existence of optimal subcodes of formally self-dual codes.

A note on the minimal binary linear code SpringerLink

A linear code of length n and dimension k is a linear subspace C with dimension k of the vector space $${\displaystyle \mathbb {F} _{q}^{n}}$$ where $${\displaystyle \mathbb {F} _{q}}$$ is the finite field with q elements. Such a code is called a q-ary code. If q = 2 or q = 3, the code is described as a binary code, or … See more In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, … See more As the first class of linear codes developed for error correction purpose, Hamming codes have been widely used in digital communication systems. For any positive integer See more Lemma (Singleton bound): Every linear [n,k,d] code C satisfies $${\displaystyle k+d\leq n+1}$$. A code C whose parameters satisfy k+d=n+1 is called … See more Some examples of linear codes include: • Repetition codes • Parity codes • Cyclic codes See more As a linear subspace of $${\displaystyle \mathbb {F} _{q}^{n}}$$, the entire code C (which may be very large) may be represented as the span of a set of $${\displaystyle k}$$ codewords (known as a basis in linear algebra). These basis codewords are … See more Codes in general are often denoted by the letter C, and a code of length n and of rank k (i.e., having k code words in its basis and k rows in its generating matrix) is generally referred … See more A code is defined to be equidistant if and only if there exists some constant d such that the distance between any two of the code's distinct codewords is equal to d. In 1984 Arrigo … See more WebNov 8, 2024 · Binary linear codes derived from vectorial Boolean functions In the past three decades, the design of binary linear codes derived from (vectorial) Boolean functions has been a research topic of increasing importance. Many classes of binary codes with good parameters have been obtained, for instance see [ 6, 17] and the references therein. riffling hitch knot

Secret sharing schemes from binary linear codes - ScienceDirect

WebIf you don't want to check all possibilities, you can use row operations to convert four columns of the matrix (of your own choosing) to the identity matrix, which will enable for very quick decoding. WebNov 2, 2024 · A pair of linear codes (C, D) of length n over $$\\mathbb {F}_q$$ F q is called a linear complementary pair (LCP) if their direct sum yields the full space $$\\mathbb {F}_q^n$$ F q n . By a result of Carlet et al. (2024), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We … WebA formula for the order of the automorphism group of a binary linear constant weight code in terms of its parameters is presented, which is a key step to determine more algebraic structures on constant weight codes with given parameters. We give a characterization for the binary linear constant weight codes by using the symmetric difference of the … rifflis meran