Information and Information and Coding TheoryCoding Theory

Hamming codes. Golay codes.Hamming codes. Golay codes.

Juris Viksna, 2015

Hamming code [7,4]

G - generator matrix

A (4 bit) message x is encoded as xG, i.e.if x = 0110 then c = xG = 0110011.

Decoding?

- there are 16 codewords, if there are no errors, we can just find the right one...- also we can note that the first 4 digits of c is the same as x :)

Other Hamming codes?

Other Hamming codes?

Sphere packing bound and perfect codes

Do perfect codes exist?

Golay codes - some historyThe brief history of the Golay Codes begins in 1949, when M. J. E. Golay published his “Notes on Digital Coding” in the Proceedings of the Institute of Electrical and Electronic Engineers”, ½ page in length. It described the (23,12,7)2 code (although he evidently did not name it after himself). This inspired a search for more perfect codes. After all, if there was some series of perfect codes, or better yet an algorithm that produces them, much of the rest of coding theory would possibly become obsolete. For any given rate and blocklength, no code with a higher minimum distance or average minimum distance can be constructed, so if it had been determined that perfect codes existed with many rates and many blocklengths, it may have been worthwhile to only search for perfect codes. It soon appeared that such prayers fell on deaf ears, as the existence of perfect codes was disproved in more and more general scenarios. Finally, in 1973, when Aimo Tietäväinen disproved the existence of perfect codes over finite fields in his “Nonexistence of Perfect Codes over Finite Fields” in the SIAM Journal of Applied Mathematics, January 1973.

Golay codesIn mathematical terms, the extended binary Golay code consists of a 12-dimensional subspace W of the space V=F2

24 of 24-bit words such that any two distinct elements of W differ in at least eight coordinates. Equivalently, any non-zero element of W has at least eight non-zero coordinates.

The possible sets of non-zero coordinates as w ranges over W are called code words. In the extended binary Golay code, all code words have Hamming weight 0, 8, 12, 16, or 24.

Up to relabelling coordinates, W is unique.

[Adapted from www.wikipedia.org]

Golay codes

010001110111100000000000

101000111011010000000000

110100011101001000000000

011010001111000100000000

101101000111000010000000

110110100011000001000000

111011010001000000100000

011101101001000000010000

001110110101000000001000

000111011011000000000100

100011101101000000000010

010001110111000000000001

G

Golay codes G24 and G23 were used by Voyager I and Voyager II to transmit color pictures of Jupiter and Saturn. Generation matrix for G24 has the form

G24 is (24,12,8) –code and the weights of all codewords are multiples of 4. G23 is obtained from G24 by deleting last symbols of each codeword of G24. G23 is (23,12,7) –code.

Golay codes

Matrix G for Golay code G24 has actually a simple and regular construction.

• The first 12 columns are formed by a unitary matrix I12, next column has all 1’s.

• Rows of the last 11 columns are cyclic permutations of the first row which has 1 at those positions that are squares modulo 11, that is

0, 1, 3, 4, 5, 9.

Ternary Golay codeThe ternary Golay code consists of 36 = 729 codewords. Its parity check matrix is:

[Adapted from www.wikipedia.org]

Any two different codewords differ in at least 5 positions. Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword. The code can also be constructed as the quadratic residue code of length 11 over the finite field F3.This is a [11,6,5] code.

Hamming codes again...

A nice idea how to correct any single error.

Can we somehow generalize this to correct a larger num,ber of errors?

Can we use the same idea to correct twoerrors?

Assume we have errors in positions i and j, we want to recoverthese positions from

[i]+[j] and f([i])+f([j]).

Can we do this?