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Asymptotically good binary code with efficient encoding & Justesen code

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### Asymptotically good binary code with efficient encoding& Justesen code

Tomer Levinboim

Error Correcting Codes Seminar (2008)

Outline

- Intro
- codes
- Singleton Bound
- Linear Codes
- Bounds
- Gilbert-Varshamov
- Hamming
- RS codes
- Code Concatention
- Examples
- Wozencraft Ensemble
- Justesen Codes

Hamming Distance

- Hamming Distance between
- The Hamming Distance is a metric
- Non negative
- Symmetric
- Triangle inequality

=

Weight

- The weight (wt) of
- Example (on board)

Code

- An (n,k,d)q code C is a function such that:
- For every

Code (parameters)

- (n,k,d)q
- Parameters
- n – block length
- k – information length
- d – minimum distance (actually, a lower bound)
- q – size of alphabet
- |C| = qk or k=logq|C|

Code (parameters div n)

- Asymptotic view of parameters as n∞:
- The rate
- Relative minimum distance
- Thus an (n,k,d)q can be written as (1,R,δ)q
- Notation: (n,k,d)q vs. [n,k,d]q – latter reserved for linear code (soon)

Trivial Code Example

- FEC3 = write each bit three time
- R = ?
- d = ?
- how many errors can we
- Detect ? (d-1)
- Correct ? t, where d=2t+1

Goal

- Would like to:
- Maximize δ – correct more
- Maximize R – send more information

* conflicting goals - would like to be able to construct an [n,k,d]q code s.t. δ>0, R>0 and both are constant.

- Minimize q – for practical reasons
- Maximize number of codewords while minimizing n and keeping d large.

Singleton Bound

- Let C be an [n,k,d]q code then
- k ≤ n – d + 1

equivalently

- R ≤ 1 – δ + o(1)
- Proof: project C to first k-1 coordinates
- On Board

Visual intuition

- On board...
- Ballq(x,r)
- r:=d
- r:=t (where d=2t+1)
- Volq(n,r) = |Ballq(x,r)|

Linear Codes

- An [n,k,d]q code C:FqKFqn is linear when:
- Fq is a field
- C is linear function (e.g., matrix)
- Linearity implies:
- C(ax+by) = aC(x) + bC(y)
- 0n member of C

Linear Codes (example)

- FEC3
- [3,1,3]2
- Hadamard – longest linear code
- [n,logn, n/2]2
- e.g., - [8,3,4]2
- (H - Matrix representation on board)
- Dimensions
- Asymptotic behavior

Linear Codes – minimum distance

- Lemma: if C:FqKFqn is linear then

Note: for clarity Cx means C(x)

- Proof:
- ≤ - trivial
- ≥ - follows from linearity (on board)

Reed-Solomon code

- Idea: oversample a polynomial
- Let q be prime power and Fq a finite field of size q.
- Let k<n and fix n elements of Fq,
- x1,x2,..xn
- Given a message m=(c0..ck-1) interpret it has the coefficients of the polynomial p

RS Codes

- Thus (c0..ck-1) is mapped to (p(x1),..p(xn))
- Linear mapping (Vandermonde)
- Using linearity, can show for x≠0

RS meet the Singleton bound

- Proof: on board
- (# of roots of a k-1 degree poly)
- Encoding time

Gilbert-Varshamov Bound Preliminaries

- Using the binary entropy we obtain
- On board

Gilbert-Varshamov Boundbound statement

- For every n and d<n/2 there is an (n,k,d)q (not necessarily linear) code such that:
- In terms of rate and relative min-distance:

Gilbert-Varshamov Bound Proof

- On Board
- Sketch of proof:
- if C is maximal then:
- And
- Now use union bound and entropy to obtain result (we show for q=2, using binary entropy)

GV-Bound

- Gilbert proved this with a greedy construction
- Varshamov proved for linear codes
- proved using random generator matrices – most matrices are good error correcting codes

Hamming Bound (Upper)

- With similar reasoning to GV bound but using
- For q=2 can show that

Bounds plot

*Madhu Sudan (Lecture 5, 2001)

Code Concatenation - Motivation

- RS codes imply we can construct good [n,k,d]q codes for any q=pk
- Practically would like to work with small q (2, 28)
- Consider the “obvious” idea for binary code generated from C – simply convert each symbol from Σn to log2q,
- What’s the problem with this approach ? (write the new code!)

Code Concatenation

- Due to Forney (1966)
- Two codes:
- Outer: Cout = [N,K,D]Q
- Inner: Cin = [n,k,d]q
- Inner code should encode each symbol of outer code k = logqQ

Code Concatenation

- What is the new code ?
- dcon = dD Proof:
- On board

Code Concatenation (Examples)

- Asymptotically
- δ = ¼
- R=logn/2n 0

Good Codes

- Can we “explicitly” build asymptotically good (linear) codes ?
- asymptotically good = constant R, δ> 0 as n∞
- Explicit = polytime constructable / logspace constructible

Asymptotically Good Codes

- GV tells us that most linear functions of a certain size are good error-correcting codes
- Can find a good code in brute-force
- Use brute force on inner-code, where the alphabet is exponentially smaller!
- Do we really need to search ?

Wozencraft Ensemble

- Lemma: There exists an ensemble of codes c1,..cN of rate ½ where N = qk-1 such that for at least (1-ε)N value of i, the code Ci has distance dis.t.
- Proof (on board), outline:
- Different codes have only 0n in common
- Let y=Cα(x), then, If wt(y)<d

y in Ball(0n, d)

there are at most Vol(n,d) “bad” codes

- For large enough n=2k, we have Vol(n,d) ≤ εN

Wozencraft Ensemble

- Implications:
- Can construct entire ensemble in O(2k)=O(2n)
- There are many such good codes, but which one do we use ?

Justesen Code

- Concatenation of:
- Cout - RS code over
- a set of inner codes
- Justesen Code: C* = Cout(C1, C2, .. CN)
- Each symbol of Cout is encoded using a different inner code Cj
- If RS has rate R C* has rate R/2

Justesen Code - δ

- Denote the outer RS code [N,K,D]Q
- Claim: C* has relative distance

Justesen Code Proof

- Intuition: like regular concatenation, but εN bad codes.
- for x≠y, the outer code induces S={j | xj≠yj},
- |S| ≥D
- There are at most εN j’s such that Cj is bad and therefore at least |S|- εN ≥ D- εN ≥ (1-R- ε)N good codes
- since RS implies D=N-(K-1)
- Each good code has relative distance ≥ d
- d* ≥ (1-R- ε)Nd

Justesen Code

- The concatenated code C* is an asymptotically good code and has a “super” explicit construction
- Can take q=2 to get such a binary code

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