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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.

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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

Asymptotically good binary code with efficient encoding& Justesen code

Tomer Levinboim

Error Correcting Codes Seminar (2008)

outline
Outline
  • Intro
    • codes
    • Singleton Bound
  • Linear Codes
  • Bounds
    • Gilbert-Varshamov
    • Hamming
  • RS codes
  • Code Concatention
    • Examples
  • Wozencraft Ensemble
  • Justesen Codes
hamming distance
Hamming Distance
  • Hamming Distance between
  • The Hamming Distance is a metric
    • Non negative
    • Symmetric
    • Triangle inequality

=

weight
Weight
  • The weight (wt) of
  • Example (on board)
slide5
Code
  • An (n,k,d)q code C is a function such that:
    • For every
code parameters
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
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
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
slide9
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
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
Visual intuition
  • On board...
  • Ballq(x,r)
    • r:=d
    • r:=t (where d=2t+1)
  • Volq(n,r) = |Ballq(x,r)|
linear codes1
Linear Codes
  • An [n,k,d]q code C:FqKFqn 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
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
Linear Codes – minimum distance
  • Lemma: if C:FqKFqn is linear then

Note: for clarity Cx means C(x)

  • Proof:
    • ≤ - trivial
    • ≥ - follows from linearity (on board)
reed solomon code
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
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
Gilbert-Varshamov Bound Preliminaries
  • Binary Entropy
  • Stirling

Implying that:

gilbert varshamov bound preliminaries1
Gilbert-Varshamov Bound Preliminaries
  • Using the binary entropy we obtain
  • On board
gilbert varshamov bound bound statement
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
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
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
singleton gv plot
Singleton / GV Plot

1

Singleton (upper)

Gilbert-Varshamov (lower)

0.5

1

hamming bound upper
Hamming Bound (Upper)
  • With similar reasoning to GV bound but using
  • For q=2 can show that
bounds plot
Bounds plot

*Madhu Sudan (Lecture 5, 2001)

code concatenation motivation
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
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 concatenation1
Code Concatenation
  • How does it work ?

* Luca Trevisan (Lecture 2)

code concatenation2
Code Concatenation
  • What is the new code ?
  • dcon = dD Proof:
    • On board
code concatenation examples
Code Concatenation (Examples)
  • Asymptotically
    • δ = ¼ 
    • R=logn/2n  0 
good codes
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 codes1
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
Wozencraft Ensemble
  • Consider the following set of codes:

such that (R=1/2) (

  • Notice that (on board)
wozencraft ensemble1
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 ensemble2
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
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 code1
Justesen Code - δ
  • Denote the outer RS code [N,K,D]Q
  • Claim: C* has relative distance
justesen code proof
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 code2
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