The minimum distance of a random linear code Jing Hao Georgia - PowerPoint PPT Presentation

The minimum distance of a random linear code Jing Hao Georgia Institute of Technology Joint work with Han Huang, Galyna Livshyts and Konstantin Tikhomirov Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Introduction on Codes Linear code: a k − dim’l subspace of F n q . wt ( c ) = |{ i | c ( i ) � = 0 }| d ( u , v ) = |{ i | u ( i ) � = v ( i ) }| Minimum distance d ( C ) = min { wt ( c ) | c ∈ C } A code with minimum distance d can correct up to d − 1 errors. 2 Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Gilbert-Varshamov Bound Theorem (Gilbert, Varshamov) For every reasonable δ and ǫ , there exists a code with rate R ≥ 1 − H q ( δ ) − ǫ and relative minimum distance δ , where H q ( x ) = x log q ( q − 1) − x log q x − (1 − x ) log q (1 − x ) Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Gilbert-Varshamov Curve Consider a random linear code C where the basis element are chosen uniformly in F n q , then every vector c ∈ C is uniform over F n q . P ( d ( C ) < d ) = P ( ∃ c ∈ C s.t. wt ( c ) < d ) � ≤ P ( wt ( c ) < d ) c ∈ C ≤ q k − n + nH q ( δ ) If we take R < 1 − H q ( δ ) − ǫ , then P ( d ( C ) < d ) ≤ q n (1 − R − H q ( δ )) < 1 There exists at least one code with desired property. Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Random Linear Codes Good error-detection Good list-decodability Applications in information theory, computer science, etc. research on random linear code is limited. Linial,Mosheiff - 2018 - gave centered moments for number of codewords with given weights. We give a full description on minimum distance of random linear code. Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Random linear codes vs random codes Ensemble 1: Random linear codes Pick k vectors { v 1 , · · · , v k } independently and uniformly random from F n q . Take span { v 1 , · · · , v k } . Let d min be the minimum distance. Ensemble 2: Random codes Pick q k vectors { Y a } a ∈ F k q indpendently and uniformly random from F n q . Forcing Y a = Y b if a // b . Let w min be the minimum distance. Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Main result Theorem (H., Huang, Livshyts, Tikhomirov) For any numbers R 1 , R 2 ∈ (0 , 1) there is c ( R 1 , R 2 , q ) > 0 with the following property. Let positive integers k , n satisfy R 1 ≤ k / n ≤ R 2 , Denote by F dmin the cumulative distribution function of d min . and F wmin be the cumulative distribution function of w min . Then | F dmin ( x ) − F wmin ( x ) | = O (exp( − c ( R 1 , R 2 , q ) √ n )) . sup x ∈ R Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Moments comparison Let C be a random linear code and Z d = |{ c ∈ C | wt ( c ) ≤ d }| Let � C be a random code and Z d = |{ c ∈ � � C | wt ( c ) ≤ d }| Proposition p ), and d 2 / n 3 / 2 ≥ C 1 ( λ 0 , p ). Suppose d , n ∈ N satisfy d n ≤ λ 0 (1 − 1 For any λ 0 ∈ (0 , 1) there are c 1 ( λ 0 , p ) > 0 and C 1 ( λ 0 , p ) > 0 such that for any positive integer m ≤ c 1 ( λ 0 , p ) d 2 / n 3 / 2 and � � − c 1 ( λ 0 , p ) d 4 p k ρ d ≥ exp , we have n 3 m m . E Z d m = (1 + O (exp( − c 1 ( λ 0 , p ) d 4 / n 3 )) + O (2 − k / 2 )) E � Z d Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Relation to density function For r > 0, let M d ( r ) = P { Z d = r } , � M d ( r ) = P { � Z d = r } � ∞ M d ( r ) r m = E Z m d r =1 � ∞ M d ( r ) r m = E � � Z m d r =1 and � ∞ P { d min ≤ d } = M d ( r ) r =1 � ∞ � P { w min ≤ d } = M d ( r ) r =1 Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Relation to density function (cont’d) Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Proof sketch Moments comparison Truncation error: � ∞ � ∞ r i � r i M d ( r ) , M d ( r ) = O (2 − h ) r = h +1 h +1 Let B = ( b ij ) where b ij = j i then B − 1 = ( b ′ ij ) satisfy ij | = O ( h − j ) | b ′ Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Proposition Let w min be the minimum of m i.i.d. binomial random variables with parameters n and p < 0 . 5 . Then the random variables γ + π w min − E w min √ � 6 Var ( w min ) converge in total variation distance to the standard Gumbel random variable with cdf e − e − x , for x ∈ R , when m and n tend to infinity simultaneously, for any fixed constant p < 0 . 5. Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Summary Random linear codes They have good error detection. Existing work on random linear codes are sparse. We give a full characterization of the density function of random linear codes by Consider a random code ensemble while forcing some of the elements to be the same. Compare the density function in these two ensembles. Show that the density function converge to Gumbel distribution. Jing Hao Georgia Institute of Technology The minimum distance of a random linear code