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On the Number of Factorizations of Polynomials over Finite Fields Rachel N. Berman Ron M. Roth CS Department Technion Haifa, Israel Berman, Roth (Technion) ISIT 2020 1 / 36 Outline Introduction 1 Lower and upper bounds 2

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On the Number of Factorizations of Polynomials over Finite Fields

Rachel N. Berman Ron M. Roth

CS Department Technion Haifa, Israel

Berman, Roth (Technion) ISIT 2020 1 / 36

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Outline

1

Introduction

2

Lower and upper bounds

3

Characterization of maximal polynomials

4

Characterization of n-maximal polynomials

5

The Average Case

Berman, Roth (Technion) ISIT 2020 2 / 36

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Notation

F = GF(q): A finite field (q a prime power). F[x]: polynomials over F. Mn: monic polynomials of degree exactly n over F. Pn: monic polynomials of degree at most n over F.

Berman, Roth (Technion) ISIT 2020 3 / 36

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A maximal polynomial

Given m ∈ Z+ and s(x) ∈ Pm:

Define:

Φ(s) = # distinct factorizations (i.e., divisors) of s(x) in Pm Υm = Υm(q) = max

s(x)∈Pm

Φ(s). Note: maximum only when degs = m. s(x) ∈ Mm is called maximal if Φ(s) = Υm.

Berman, Roth (Technion) ISIT 2020 4 / 36

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Coding application

Upper bound over GF(2), [Piret‘84]: Υm(2) ≤ (81/16)(m/log2 m)(1+om(1)). Application: most binary shortened cyclic codes approach the Gilbert–Varshamov bound.

Berman, Roth (Technion) ISIT 2020 5 / 36

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Coding application

Upper bound over GF(2), [Piret‘84]: Υm(2) ≤ (81/16)(m/log2 m)(1+om(1)). Application: most binary shortened cyclic codes approach the Gilbert–Varshamov bound.

We improve bounds for any q:

Υm(q) = 2(m/logq m)(1±om(1)) where om(1) − − − →

m→∞ 0.

Berman, Roth (Technion) ISIT 2020 5 / 36

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An n-maximal polynomial

Given n ∈ Z+ and s(x) ∈ P2n:

(n,n)-factorization:

An (n,n)-factorization of s(x) is an ordered pair (u(x),v(x)) ∈ Pn ×Pn s.t. s(x) = u(x)·v(x).

Define:

Φn(s) = Φn,n(s) = # distinct (n,n)-factorizations of s(x) Υn,n = Υn,n(q) = max

s(x)∈P2n

Φn(s). s(x) ∈ P2n is called n-maximal if Φn(s) = Υn,n.

Berman, Roth (Technion) ISIT 2020 6 / 36

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Application: list decoding of rank-metric codes

A construction: diagonally-interleaved codes [Roth‘91]. List decoder for D-I codes for: ([Roth’17])

Array size: (n +1)×(n +1) Minimum rank distance: 2 Decoding radius: 1

Υn,n = largest list size. Over large fields (|F| ≥ 2n−1): list size = 4n−on(1) [Roth’17].

Berman, Roth (Technion) ISIT 2020 7 / 36

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Application: list decoding of rank-metric codes

A construction: diagonally-interleaved codes [Roth‘91]. List decoder for D-I codes for: ([Roth’17])

Array size: (n +1)×(n +1) Minimum rank distance: 2 Decoding radius: 1

Υn,n = largest list size. Over large fields (|F| ≥ 2n−1): list size = 4n−on(1) [Roth’17].

We show:

Over small fields list size is sub-exponential. Average list size: linear; E{“list size”} = (n +1)(1+O(1/q)) . Variance: polynomial; Var{“list size”} = O(n4) .

Berman, Roth (Technion) ISIT 2020 7 / 36

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Two related combinatorial problems

Problem 1 — Ordinary factorization

Given m ∈ Z+, compute Υm and characterize the maximal polynomials in Mm.

Problem 2 — (n,n)-factorization

Given n ∈ Z+, compute Υn,n and characterize the n-maximal polynomials in P2n.

Berman, Roth (Technion) ISIT 2020 8 / 36

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Lower and Upper Bounds on Υm and Υn,n

Berman, Roth (Technion) ISIT 2020 9 / 36

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Bounds

Note: For all s(x) ∈ P2n we have Φn(s) ≤ Φ(s), thus Υn,n ≤ Υ2n ≤ Υ2n+1

Theorem (Upper bound on Υm)

For all m ∈ Z+: log2 Υm ≤ m logq m ·

  • 1+O

logq logq m logq m

  • .

Theorem (Lower bound on Υn,n)

For all n ∈ Z+: log2 Υn,n ≥ 2n logq n ·

  • 1−O
  • 1

logq n

  • .

Berman, Roth (Technion) ISIT 2020 10 / 36

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Bounds (continued)

We conclude Υm = 2(m/logq m)(1±om(1)) Υn,n = 2(2n/logq n)(1±on(1)) where om(1) − − − →

m→∞ 0.

Berman, Roth (Technion) ISIT 2020 11 / 36

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Characterization of Maximal Polynomials

Berman, Roth (Technion) ISIT 2020 12 / 36

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Representing a polynomial by a histogram

Monic irreducible polynomials over F: (pi(x))∞

i=1

Assume: degpi(x) ≤ degpi+1(x).

Example

For F = GF(2): p1(x) = x, p3(x) = x2 +x +1, p5(x) = x3 +x2 +1, p2(x) = x +1, p4(x) = x3 +x +1, p6(x) = x4 +x +1... Define di = degpi(x).

Berman, Roth (Technion) ISIT 2020 13 / 36

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Representing a polynomial by a histogram (continued)

Given a polynomial s(x) ∈ F[x]: Factorize s(x): s(x) =

t

∏

i=1

pi(x)ri ri ≥ 0, rt > 0 Histogram of s(x): r(s) = (r1 r2 ... rt) Define ρ(s) = max

i∈Z+ :di=1ri = q

max

i=1 ri.

Example:

Berman, Roth (Technion) ISIT 2020 14 / 36

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s(x) = x3 ·(x +1)2 ·(x2 +x +1)·(x3 +x2 +1)·(x4 +x +1)

1 x 1 x +1 2 x2 +x +1 3 x3 +x +1 3 x3 +x2 +1 4 x4 +x +1 4 4 5 1 2 3 4 5 di : ri

r(s) = (3,2,1,0,1,1) ρ(s) = 3 Pi[x]:

Berman, Roth (Technion) ISIT 2020 15 / 36

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Maximal polynomial for m = 180

1 1 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10

di: ri

Berman, Roth (Technion) ISIT 2020 16 / 36

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Explicit formula for Φ(s)

It is easy to see that Φ(s) =

t

∏

i=1

(ri +1)

Berman, Roth (Technion) ISIT 2020 17 / 36

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Explicit formula for Φ(s)

It is easy to see that Φ(s) =

t

∏

i=1

(ri +1) = ⇒ degs = m.

Berman, Roth (Technion) ISIT 2020 17 / 36

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Structural properties of maximal polynomials

Histogram is monotonically decreasing. Histogram is all-positive.

Berman, Roth (Technion) ISIT 2020 18 / 36

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Structural properties of maximal polynomials

Histogram is monotonically decreasing. Histogram is all-positive.

General proof method

Assume s ∈ Pm does not satisfy the property. Construct from s a polynomial ˜ s ∈ Pm for which Φ(˜ s) > Φ(s). Conclude that s cannot be maximal.

Berman, Roth (Technion) ISIT 2020 18 / 36

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Structural properties of maximal polynomials (continued)

Degree vs. multiplicity of an irreducible factor

For every i ∈ [1 : t]: ρ +1 ri +2 ≤ di < ρ +1 ri . Equivalently: ri ∈ {⌊ρ/di⌋, ⌊ρ/di⌋−1}.

Maximal degree of irreducible factor

  • logq(m/8)
  • < dt ≤ dt+1 ≤
  • logq m
  • +1.

Estimation of ρ

ρ = logq m ln2 ±O

  • logq logq m
  • .

Berman, Roth (Technion) ISIT 2020 19 / 36

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Maximal polynomial for m = 180

1 1 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10

di: ri

log2(180) = 7.491 dt = 7 log2(180)/ln2 = 10.808 ρ = 9

Berman, Roth (Technion) ISIT 2020 20 / 36

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Improved characterization for large di

Theorem

For every i ∈ [1 : t] such that di ≥ Θ

  • logq logq m
  • :

log2

  • 1+

1 ri+1

  • ·
  • logq m
  • −O(1)

< di ≤ log2

  • 1+ 1

ri

  • ·
  • logq m
  • +O(1).

Equivalently: ri =

  • 1
  • 2(di±O(1))/⌊logq m⌋ −1
  • .

Berman, Roth (Technion) ISIT 2020 21 / 36

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“Long tail” of 1-s

ri > 1 only when di logq m < log2(3/2)+om(1) ≈ 0.585. Recall dt ≈ logq m.

Conclusion

For a given q and m → ∞, almost all of the multiplicities in r(s) are 1.

Berman, Roth (Technion) ISIT 2020 22 / 36

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Characterization of n-Maximal Polynomials

Berman, Roth (Technion) ISIT 2020 23 / 36

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Reminder: n-maximal polynomial

Given n ∈ Z+ and a polynomial s(x) ∈ P2n (i.e. degs ≤ 2n):

(n,n)-factorization:

An (n,n)-factorization of s(x) is an ordered pair (u(x),v(x)) ∈ Pn ×Pn s.t. s(x) = u(x)·v(x). s(x) ∈ P2n is n-maximal if s(x) has maximal # of (n,n)-factorizations.

Berman, Roth (Technion) ISIT 2020 24 / 36

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Notation

For n ∈ Z+ and s(x) = ∏t

i=1 pi(x)ri ∈ P2n:

r0 = 2n −degs d0 ≡ 1 rn(s) = (r0 r(s)) = (ri)t

i=0

ρn(s) = max{r0,ρ(s)} = maxi∈Z≥0 :di=1 ri

Berman, Roth (Technion) ISIT 2020 25 / 36

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s(x) = x ·(x +1)3 ·(x2 +x +1)2 ·(x3 +x +1) n = 7, 2n = 14, degs = 11

1 x 1 x +1 2 x2 +x +1 3 x3 +x +1 3 x3 +x2 +1 4 x4 +x +1 4 4 1 2 3 4 di : ri

r(s) = (1,3,2,1) Pi[x]:

Berman, Roth (Technion) ISIT 2020 26 / 36

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s(x) = x ·(x +1)3 ·(x2 +x +1)2 ·(x3 +x +1) n = 7, 2n = 14, degs = 11, r0 = 3

1 y 1 x 1 x +1 2 x2 +x +1 3 x3 +x +1 3 x3 +x2 +1 4 x4 +x +1 4 1 2 3 4 di : ri

rn(s) = (3,1,3,2,1) ρn(s) = 3 Pi[x]:

Berman, Roth (Technion) ISIT 2020 27 / 36

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Results

Same structural properties as for maximal polynomials, where we replace: m ← → 2n r(s) ← → rn(s) ρ(s) ← → ρn(s) 1 ≤ i ≤ t ← → 0 ≤ i ≤ t.

Berman, Roth (Technion) ISIT 2020 28 / 36

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Results

Same structural properties as for maximal polynomials, where we replace: m ← → 2n r(s) ← → rn(s) ρ(s) ← → ρn(s) 1 ≤ i ≤ t ← → 0 ≤ i ≤ t. Ordinary factorizations: Φ(s) =

t

∏

i=1

(ri +1). (n,n)-factorizations: Φn(s) =??? Therefore, here we need more intricate proofs.

Berman, Roth (Technion) ISIT 2020 28 / 36

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90-maximal polynomial

1 1 1 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10

di: ri

Berman, Roth (Technion) ISIT 2020 29 / 36

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Difference between n-maximal and maximal polynomials

It follows that r0 = Θ(logq n), thus degs = 2n −Θ(logq n) < 2n.

Berman, Roth (Technion) ISIT 2020 30 / 36

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Difference between n-maximal and maximal polynomials

It follows that r0 = Θ(logq n), thus degs = 2n −Θ(logq n) < 2n. In contrast, given m ∈ Z+, any maximal polynomial has degree exactly m.

Berman, Roth (Technion) ISIT 2020 30 / 36

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90-maximal polynomial

1 1 1 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10

di: ri

Berman, Roth (Technion) ISIT 2020 31 / 36

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Maximal polynomial for m = 180

1 1 1 2 3 3 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 1 2 3 4 5 6 7 8 9 10

di: ri

Berman, Roth (Technion) ISIT 2020 32 / 36

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The Average Case

Berman, Roth (Technion) ISIT 2020 33 / 36

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Average case — ordinary factorizations

Given m ∈ Z+: Sample space: Mm with uniform distribution. Random variable Φm over s(x) ∈ Mm: Φm : s → Φ(s).

Theorem

E{Φm} = m +1 and Var{Φm} = q −1 q m +1 3

  • .

Berman, Roth (Technion) ISIT 2020 34 / 36

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Average case — ordinary factorizations

Given m ∈ Z+: Sample space: Mm with uniform distribution. Random variable Φm over s(x) ∈ Mm: Φm : s → Φ(s).

Theorem

E{Φm} = m +1 and Var{Φm} = q −1 q m +1 3

  • .

From Markov and Chebyshev inequalities: Prob

  • Φm ≥ m1+ε

≤ O

  • m−max{ε,2ε−1}

∀ ε > 0.

Conclusion

Prob{Φm being super-linear in m} − − − →

m→∞ 0

Berman, Roth (Technion) ISIT 2020 34 / 36

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Proposition

For any (fixed) ε > 0, Prob

  • Φm ≥ mε+ln2

≤ O

  • m−κ(ε)

, where κ(ε) > 0. The proof uses the Chernoff bound. Conclusion: The median of Φm is sub-linear in m.

Berman, Roth (Technion) ISIT 2020 35 / 36

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Average case — (n,n)-factorizations

Given n ∈ Z+: Sample space: P2

n = Pn ×Pn with uniform distribution.

Random variable Φn,n over (u,v) ∈ P2

n:

Φn,n : (u,v) → Φn(u ·v) (i.e., # of (n,n)-factorizations of the product u ·v).

Theorem

E{Φn,n} = (n +1)(1+O(1/q)) and Var{Φn,n} = O(n4).

Conclusion

Prob{Φn,n being super-linear in n} − − − →

n→∞ 0

Berman, Roth (Technion) ISIT 2020 36 / 36