Deletion Decoding Codes from GRS Codes L McAven, R Safavi-Naini, Y - - PowerPoint PPT Presentation

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Jul 27, 2023 440 likes •542 views

Deletion Decoding Codes from GRS Codes L McAven, R Safavi-Naini, Y Wang CIS- UoW AUSTRALIA Motivation 1 2 3 4 4 1 2 3 2 1 3 4 Tracing shortened fingerprints Deletion Correction Codewords, of length n A received word has n-r

SLIDE 1

Deletion Decoding Codes from GRS Codes

L McAven, R Safavi-Naini, Y Wang CIS- UoW AUSTRALIA

SLIDE 2

Motivation

Tracing shortened fingerprints

1 2 2 2 1 1 3 4 4 3 3 4

SLIDE 3

Deletion Correction

Codewords, of length n
A received word has n-r elements, order

preserved.

Problem: Recover the original word
Transmit x=(1 6 2 5); receive y=(1 6 5).
Applications: Synchronisation, traitor tracing.
A code can correct r deletions if words of

length n-r are subwords of at most one codeword.

SLIDE 4

Example

5362 4251 3140 2036 1625 0514 6403 4065 3654 2543 1432 0321 6210 5106 3461 2350 1246 0135 6024 5613 4502 2164 1053 0642 6531 5420 4316 3205 1560 0456 6345 5234 4123 3012 2601 0263 6152 5041 4630 3526 2415 1304 6666 5555 4444 3333 2222 1111 0000

SLIDE 5

Deletion Correcting Codes

Constructions

Perfect code

Combinatorial structures

No efficient decoding

Decoding: Brute force:

For a substring x of length n-r, find codewords that contain x Repeat for all x

SLIDE 6

Generalised Reed-Solomon

Generalised Reed-Solomon codes are known for

their error correcting properties.

Let Γ be a GRS code GRS(k,q,n,α,v).

A codeword c is obtained from a polynomial fc over

GF(q), of degree ≤ k,

Evaluate polynomial at a subset of points of GF(q) There are q^(k+1) codewords.

Theorem: There exist GRS(k,q,n,α,v) codes capable of correcting deletions.

SLIDE 7

Decoding shortened words using list decoding

Efficient list decoding algorithm for GRS codes:

Guruswami and Sudan (1999).

Applicable to deletion decoding.

Safavi-Naini and Wang ( ACM DRM 2002).

Let t=(n-r) length of the received word. Then for

n≥log q and n>k(r+1)+r the decoding algorithm has running time;

}) , max{ (

3 6 2 6 3

1 ) ( ) ( 6 k kn t n k

t O

−

SLIDE 8

Deletion capacity of GRS codes

Exhaustive searches over small fields (q<~7). Partial searches over fields (~7<q<~149). Correct over half the code length. Tabulate length of unique substrings for codelength n