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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

• f Fn

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

2

errors.

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 − Hq(δ) − ǫ and relative minimum distance δ, where Hq(x) = x logq(q − 1) − x logq x − (1 − x) logq(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 Fn

q, then every vector c ∈ C is uniform over Fn q.

P(d(C) < d) = P(∃c ∈ C s.t. wt(c) < d) ≤

• c∈C

P(wt(c) < d) ≤ qk−n+nHq(δ) If we take R < 1 − Hq(δ) − ǫ, then P(d(C) < d) ≤ qn(1−R−Hq(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 {v1, · · · , vk} independently and uniformly random from Fn

q.

Take span{v1, · · · , vk}. Let dmin be the minimum distance.

Ensemble 2: Random codes

Pick qk vectors {Ya}a∈Fk

q indpendently and uniformly random

from Fn

q.

Forcing Ya = Yb if a//b. Let wmin 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 R1, R2 ∈ (0, 1) there is c(R1, R2, q) > 0 with the following property. Let positive integers k, n satisfy R1 ≤ k/n ≤ R2, Denote by Fdmin the cumulative distribution function of dmin. and Fwmin be the cumulative distribution function of wmin. Then sup

x∈R

|Fdmin(x) − Fwmin(x)| = O(exp(−c(R1, R2, q) √n)).

Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Moments comparison

Let C be a random linear code and Zd = |{c ∈ C|wt(c) ≤ d}| Let C be a random code and

• Zd = |{c ∈

C|wt(c) ≤ d}| Proposition Suppose d, n ∈ N satisfy d

n ≤ λ0(1 − 1 p), and d2/n3/2 ≥ C1(λ0, p).

For any λ0 ∈ (0, 1) there are c1(λ0, p) > 0 and C1(λ0, p) > 0 such that for any positive integer m ≤ c1(λ0, p)d2/n3/2 and pkρd ≥ exp

• − c1(λ0,p)d4

n3m

• , we have

EZd m = (1 + O(exp(−c1(λ0, p)d4/n3)) + O(2−k/2)) E Zd

m.

Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Relation to density function

For r > 0, let Md(r) = P{Zd = r} , Md(r) = P{ Zd = r}

∞

• r=1

Md(r)rm = EZ m

d ∞

• r=1
• Md(r)rm = E

Z m

d

and P{dmin ≤ d} =

∞

• r=1

Md(r) P{wmin ≤ d} =

∞

• r=1
• Md(r)

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=h+1

riMd(r) ,

∞

• h+1

ri Md(r) = O(2−h) Let B = (bij) where bij = ji then B−1 = (b′

ij) satisfy

|b′

ij| = O(h−j)

Jing Hao Georgia Institute of Technology The minimum distance of a random linear code

Proposition Let wmin be the minimum of m i.i.d. binomial random variables with parameters n and p < 0.5. Then the random variables γ + π √ 6 wmin − Ewmin

• Var(wmin)

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