Lecture-7

Distance properties of Block Codes (cond..)

Minimum Distance Decoding

Some bounds on the Code size

A block code can be specified by (n, k, dmin)

• Fact: The minimum distance of a code

determines its error-detecting ability and

error-correcting ability.

• d ↑ (Good Error correction capability)

but k / n ↓ (Low rate)

Error detection

• Let c be the transmitted codeword and let r bethe received n-tuple. We call the distance d(c; r)the weight of the error.

• let c1; c2 be two closest code words. If c1 istransmitted but c2 is received, then an error ofweight dmin has occurred that cannot bedetected.

• If the no. of errors < dmin , syndrome will havenon zero weight – error detection possible.

• (n, k, dmin) LBC can detect (dmin –1) number oferrors per code word.

Error correction

• A block code is a set of M vectors in an n-

dimensional space with geometry defined

by Hamming distance.

• The optimal decoding procedure is often

nearest-neighbour decoding: the received

vector r is decoded to the nearest

codeword.

Error correction-contd…

Contd….

Error correction-contd…

• Code words-center of a sphere of radius t`

• The “spheres" of radius t` surrounding the codewords do not overlap.

• When t` errors occur, the decoder can unambiguously decide which codeword was transmitted.

• Using nearest-neighbour decoding, errors of weight t` can be corrected if and only if

(2t` + 1) dmin –1) .

Error correction-contd…

• (n, k, dmin) LBC can detect (dmin –1) number

of errors and correct up to (dmin –1)/2 per

code word.

• In general ed + ec dmin –1.

Singleton bound

Another Proof:

Write G=[ Ik P]

Ik contributes 1 to wmin.

P contributes at most n - k to w min.

Hence the bound is satisfied

Hamming Codes

• Problem: Design a LBC with dmin 3 for

some block length n = 2m - 1.

• If dmin = 3, then every pair of columns of H

is independent.

• i.e., for binary code, this requires only that

• – no two columns are equal;

• – all columns are nonzero.

Hamming codes-contd…

• Let the H matrix has m rows. Each column

is an m-bit number.There are 2m -1

possible columns. This defines a (2m –1,

2m –1-m) code.

• The minimum weight is 3 and the code

can correct single error per code word.

New Codes from Existing Codes

New Codes from Existing Codes

How?

• Adding a check symbol expands a code.

• Adding an info symbol lengthens a code.

• Dropping a check symbol punctures a

code.

• Dropping an info symbol shortens a code.

Expansion

• Consider a binary (n; k) code with odd minimum distance dmin .

• Add one additional position which checks (even) parity on all n positions.

• The dimension k of the code is unchanged.

• – dmin increases by one.

• – The code length n increases by one.

• As an example of an expanded code, consider

• (2m, 2m –1- m) Hamming code with dmin = 4: