Discussion part for experiment properties of alkenesFull description
Concrete Tests and Analysis on site, Concrete Design Mix, Concrete Slump TestFull description
This lecture slides describes the properties of dry gas from the reservoirFull description
VCI Prescribed syllabusFull description
Full description
Properties of Lubricating Oil
This document briefly explains properties of aggregate.Full description
Carter & BentleyDescripción completa
hc verma Hindi
Class 11 CBSE NotesFull description
Full description
Full description
Full description
ASM - Aluminium Alloys
ASM - Aluminium Alloys
Mechanical Properties of PolymersFull description
Full description
Magical workings of woodDescripción completa
Discussion part for experiment properties of alkenesFull description
mekanika reservoar
Descripción completa
Lecture-7
Distance
properties of Block Codes (cond..)
Minimum
Some
Distance Decoding
bounds on the Code size
A block code can be specified by (n, k , d min) •
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 be the received n-tuple. We call the distance d(c; r) the weight of the error. • let c1; c2 be two closest code words. If c 1 is transmitted but c2 is received, then an error of weight dmin has occurred that cannot be detected. • If the no. of errors < dmin , syndrome will have non zero weight – error detection possible. • (n, k , d min) LBC can detect (dmin –1) number of errors per code word. •
Error correction • A
block code is a set of M vectors in an ndimensional 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 code words 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 , d min) LBC can detect (dmin –1) number
of errors and correct up to (d min –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 d min 3 for some block length n = 2m - 1.
•
If d min = 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 2 m -1 possible columns. This defines a (2 m –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 d min . Add one additional position which checks (even) parity on all n positions. The dimension k of the code is unchanged. – d min 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 d min = 4: