Block mark

The block mark (block ふごう British: Block code) is a kind of the transmission channel mark in the mark theory. I provide an information rate (the thing which expressed quantity of the transfer information per one second in a bit) not to go over the channel capacity while enabling decoding error-free as possible on the reception side by adding long-windedness to a message.

The characteristic of the block mark is the point that is the mark of the fixed length and is different from the source of information mark such as the Huffman mark and the transmission channel mark such as the convolutional code. Generally, I assume information word of the k figure input and generate mark word of the n figure.

The block mark was used as transmission channel mark with an early cell-phone.

Table of contents

Form definition

The block mark is the alphabet ${\displaystyle S}$  Encode で character string to be constructed, and the mark word ${\displaystyle S}$  There is it on every each inner の letter.${\displaystyle (k_{1},k_{2},\ldots ,k_{m})}$  を ${\displaystyle |S|}$  I assume it a row of under natural number.${\displaystyle S={s_{1},s_{2},\ldots ,s_{n}}}$  とし, a certain word ${\displaystyle W}$  の spelling ${\displaystyle W=s_{k_{1}}s_{k_{2}}\ldots s_{k_{m}}}$  であるとき,${\displaystyle W}$  I became を mark ${\displaystyle C(W)}$  It becomes like は next.

${\displaystyle C(W)=C(s_{k_{1}})C(s_{k_{2}})\ldots C(s_{k_{m}})}$ 

A[n,d]

The greatest mark number of words when I fixed length and ability for correction (expressed in Hamming distance d) of the mark word as a thing indicating the trade-offs of efficiency (transfer rate) and the ability for correction is used. I describe the biggest mark number of words in case of mark word length n and Hamming distance d with A[n,d].

Information rate

Binary block mark ${\displaystyle C}$  の mark number of words ${\displaystyle A}$ When I assumed, mark word length n,${\displaystyle C}$  The の information rate is defined as follows.

${\displaystyle {\frac {\!log_{2}(A)}{n}}}$ 

In the case of an independent information bit, the information rate is as follows k bit among mark word.

${\displaystyle {\frac {\!log_{2}(2^{k})}{n}}={\frac {k}{n}}}$ 

Ball filling

The block mark is related to ball filling closely. It is easy to visualize two dimensions. I put several same coins on the table and become flat. Then an alveolate hexagonal pattern appears. But it is higher in the level of the block mark and cannot easily visualize it. By the mark theory, I employ N dimension ball model. For example, the ゴレイ mark used for communication in the outer space is based on 24 dimensions of ball filling. In the case of the binary mark, this dimension is the same as above-mentioned mark word length.

Outside link

• It is Takeshi Hara (Saga University), "information algebra and mark theory" lecture document 9 in a block mark
• Mathematical model Mitsuru Kawazoe (Kyoto University) of the block mark

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This article is taken from the Japanese Wikipedia Block mark

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