Contents

Block code: Difference between revisions

Revision as of 13:38, 26 August 2009

Block coding was the primary type of channel coding used in earlier mobile communication systems.

Formal definition

A block code is a code which encodes strings formed from an alphabet set ${\displaystyle S}$ into code words by encoding each letter of ${\displaystyle S}$ separately. Let ${\displaystyle (k_{1},k_{2},\ldots ,k_{m})}$ be a sequence of natural numbers each less than ${\displaystyle |S|}$. If ${\displaystyle S={s_{1},s_{2},\ldots ,s_{n}}}$ and a particular word ${\displaystyle W}$ is written as ${\displaystyle W=s_{k_{1}}s_{k_{2}}\ldots s_{k_{m}}}$, then the code word corresponding to ${\displaystyle W}$, namely ${\displaystyle C($W)} , is

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

A[n,d]

The trade-off between efficiency (large code rate) and correction capabilities can also be seen from the attempt to, given a fixed codeword length and a fixed correction capability (represented by the Hamming distanced) maximize the total amount of codewords. A[n,d] is the maximum number of codewords for a given codeword length n and Hamming distance d.

Information rate

When ${\displaystyle C}$ is a binary block code, consisting of ${\displaystyle A}$ codewords of length n bits, then the code rateof${\displaystyle C}$ is defined as

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

When if the first k bits of a codeword are independent information bits, then the code rate is

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

Sphere packings and lattices

Block codes are tied to the sphere packing problem which has received some attention over the years. In two dimensions, it is easy to visualize. Take a bunch of pennies flat on the table and push them together. The result is a hexagon pattern like a bee's nest. But block codes rely on more dimensions which cannot easily be visualized. The powerful Golay code used in deep space communications uses 24 dimensions. If used as a binary code (which it usually is,) the dimensions refer to the length of the codeword as defined above.

The theory of coding uses the N-dimensional sphere model. For example, how many pennies can be packed into a circle on a tabletop or in 3 dimensions, how many marbles can be packed into a globe. Other considerations enter the choice of a code. For example, hexagon packing into the constraint of a rectangular box will leave empty space at the corners. As the dimensions get larger, the percentage of empty space grows smaller. But at certain dimensions, the packing uses all the space and these codes are the so called perfect codes. There are very few of these codes.

Another item which is often overlooked is the number of neighbors a single codeword may have. Again, let's use pennies as an example. First we pack the pennies in a rectangular grid. Each penny will have 4 near neighbors (and 4 at the corners which are farther away). In a hexagon, each penny will have 6 near neighbors. Respectively, in three and four dimensions, the maximum packing is given by the 12-face and 24-cell with 12 and 24 neighbors, respectively. When we increase the dimensions, the number of near neighbors increases very rapidly. In general, the value is given by the kissing numbers.

The result is the number of ways for noise to make the receiver choose a neighbor (hence an error) grows as well. This is a fundamental limitation of block codes, and indeed all codes. It may be harder to cause an error to a single neighbor, but the number of neighbors can be large enough so the total error probability actually suffers.

References

This article needs additional citations for verification. Please help improve this articlebyadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Block code" – news · newspapers · books · scholar · JSTOR (September 2008) (Learn how and when to remove this message)

• J.H. van Lint (1992). Introduction to Coding Theory. GTM. Vol. 86 (2nd ed ed.). Springer-Verlag. p. 31. ISBN 3-540-54894-7. {{cite book}}: |edition= has extra text (help)
• F.J. MacWilliams (1977). The Theory of Error-Correcting Codes. North-Holland. p. 35. ISBN 0-444-85193-3. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Coding Concepts and Block Coding