Non-binary Hamming codes? - Mathematics Stack Exchange
2023年2月6日 · 2. This document contains some information on non-binary Hamming codes. The formal definition is on page 51. I'm not sure about nonlinear Hamming codes. If your (linear) Hamming code is C C, you could form a nonlinear code C′ C ′ by taking some x ∉ C x ∉ C and definining C′ = x + C C ′ = x + C. However, I would hesitate to call that ...
Efficient computation of the minimum distance of a binary linear …
I know the method that compute all the codewords and take minimum non-zero weight code will be the minimum distance. but it's an exponential time algorithm. Is there any efficient algorithm to do the same?
Binary linear codes and weights - Mathematics Stack Exchange
1. Let E ⊂ C be the code containing all the codewords of even weight in C. In order to show that the code is linear, we only need to show that it is closed under addition. In other words, we need to show that if a, b ∈ E, then a + b ∈ E. Note that a + b surely is in the original code C, because C is linear. This means that it is ...
Non-Systematic Binary Linear Codes - Mathematics Stack Exchange
2023年1月3日 · Any linear map T: {0, 1}k → {0, 1}n T: {0, 1} k → {0, 1} n where n = k + r n = k + r defines a linear code. To obtain a good code, you need more properties of this map, of course. So, yes a non-systematic code need not necessarily contain the k k bits as some subset of the codeword coordinates. Share.
Proving that there doesn't exist a binary code with parameters [6,5,4]
2019年3月30日 · I'm new to code theory and I'm having trouble with proving if a code doesn't exist in general. I found a binary code with the parameters [6,4,4] but I don't really understand why there cant be one with parameters [6,5,4]. It would help a lot having an example of such problems. Thanks in advance.
Why the only binary MDS codes are trivial ones?
2018年5月10日 · $\begingroup$ +1 But I feel like pointing out that the Hamming code does not usually achieve the Singleton bound. The length three Hamming code is equal to the repetition code, and that is the only Hamming code satisfying the Singleton bound. Which really shows how useless the Singleton bound is in the binary world.
How is a Generator Matrix for a (7, 4) Hamming code created?
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Weights of Binary Linear Code - Mathematics Stack Exchange
2017年4月13日 · Partition C C into two subsets C0 C 0 and C1 C 1 consisting respectively of all the codewords of even Hamming weight and all the codewords of odd Hamming weight. Show that for every linear binary code, C0 C 0 is a non-empty set. Hint: find a codeword of even weight in C C. (Subhint: 0 0 is an even integer).
Finding the parity check matrix for $(15, 11)$ Hamming Codes
2013年4月12日 · By duality, this is a check matrix of a binary $(15,11)$ Hamming code. For general alphabets $\mathbb{F}_q$, you have to select a system of projective representatives of the non-zero vectors in $\mathbb{F}_q^k$ for the columns.
information theory - Word lengths of optimal binary code
2016年2月1日 · An optimal code minimises expected word length across all decipherable codes, but it doesn't necessarily minimise the sum of the word lengths. It depends on the probabilites of the messages. For example, doing Huffman coding on the probabilites 0.7, 0.1, 0.1, 0.1 gives codewords 0, 11, 100, 101. This is optimal, but the sum of the codeword ...