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 Characterization of maximal polynomials 3 Characterization of n -maximal polynomials 4 The Average Case 5 Berman, Roth (Technion) ISIT 2020 2 / 36
Notation F = GF ( q ): A finite field ( q a prime power). F [ x ]: polynomials over F . M n : monic polynomials of degree exactly n over F . P n : monic polynomials of degree at most n over F . Berman, Roth (Technion) ISIT 2020 3 / 36
A maximal polynomial Given m ∈ Z + and s ( x ) ∈ P m : Define: Φ( s ) = # distinct factorizations (i.e., divisors) of s(x) in P m Υ m = Υ m ( q ) = max Φ( s ) . s ( x ) ∈ P m Note: maximum only when deg s = m . s ( x ) ∈ M m is called maximal if Φ( s ) = Υ m . Berman, Roth (Technion) ISIT 2020 4 / 36
Coding application Upper bound over GF (2), [Piret‘84]: Υ m (2) ≤ (81 / 16) ( m / log 2 m )(1+ o m (1)) . Application: most binary shortened cyclic codes approach the Gilbert–Varshamov bound. Berman, Roth (Technion) ISIT 2020 5 / 36
Coding application Upper bound over GF (2), [Piret‘84]: Υ m (2) ≤ (81 / 16) ( m / log 2 m )(1+ o m (1)) . Application: most binary shortened cyclic codes approach the Gilbert–Varshamov bound. We improve bounds for any q : Υ m ( q ) = 2 ( m / log q m )(1 ± o m (1)) where o m (1) − m → ∞ 0. − − → Berman, Roth (Technion) ISIT 2020 5 / 36
An n -maximal polynomial Given n ∈ Z + and s ( x ) ∈ P 2 n : ( n , n )-factorization: An ( n , n ) -factorization of s ( x ) is an ordered pair ( u ( x ) , v ( x )) ∈ P n × P n s.t. s ( x ) = u ( x ) · v ( x ) . Define: Φ n ( s ) = Φ n , n ( s ) = # distinct (n,n)-factorizations of s(x) Υ n , n = Υ n , n ( q ) = max Φ n ( s ) . s ( x ) ∈ P 2 n s ( x ) ∈ P 2 n is called n -maximal if Φ n ( s ) = Υ n , n . Berman, Roth (Technion) ISIT 2020 6 / 36
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 Υ n , n = largest list size. Over large fields ( | F | ≥ 2 n − 1): list size = 4 n − o n (1) [Roth’17]. Berman, Roth (Technion) ISIT 2020 7 / 36
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 Υ n , n = largest list size. Over large fields ( | F | ≥ 2 n − 1): list size = 4 n − o n (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 ( n 4 ) . Berman, Roth (Technion) ISIT 2020 7 / 36
Two related combinatorial problems Problem 1 — Ordinary factorization Given m ∈ Z + , compute Υ m and characterize the maximal polynomials in M m . Problem 2 — ( n , n )-factorization Given n ∈ Z + , compute Υ n , n and characterize the n -maximal polynomials in P 2 n . Berman, Roth (Technion) ISIT 2020 8 / 36
Lower and Upper Bounds on Υ m and Υ n , n Berman, Roth (Technion) ISIT 2020 9 / 36
Bounds Note: For all s ( x ) ∈ P 2 n we have Φ n ( s ) ≤ Φ( s ), thus Υ n , n ≤ Υ 2 n ≤ Υ 2 n +1 Theorem (Upper bound on Υ m ) For all m ∈ Z + : � � log q log q m �� m log 2 Υ m ≤ log q m · 1+ O . log q m Theorem (Lower bound on Υ n , n ) For all n ∈ Z + : � � �� 2 n 1 log 2 Υ n , n ≥ log q n · 1 − O . log q n Berman, Roth (Technion) ISIT 2020 10 / 36
Bounds (continued) We conclude Υ m = 2 ( m / log q m )(1 ± o m (1)) Υ n , n = 2 (2 n / log q n )(1 ± o n (1)) where o m (1) − m → ∞ 0. − − → Berman, Roth (Technion) ISIT 2020 11 / 36
Characterization of Maximal Polynomials Berman, Roth (Technion) ISIT 2020 12 / 36
Representing a polynomial by a histogram Monic irreducible polynomials over F : ( p i ( x )) ∞ i =1 Assume: deg p i ( x ) ≤ deg p i +1 ( x ) . Example For F = GF (2) : p 3 ( x ) = x 2 + x +1 , p 5 ( x ) = x 3 + x 2 +1 , p 1 ( x ) = x , p 4 ( x ) = x 3 + x +1 , p 6 ( x ) = x 4 + x +1 ... p 2 ( x ) = x +1 , Define d i = deg p i ( x ). Berman, Roth (Technion) ISIT 2020 13 / 36
Representing a polynomial by a histogram (continued) Given a polynomial s ( x ) ∈ F [ x ]: Factorize s ( x ): t p i ( x ) r i ∏ s ( x ) = r i ≥ 0 , r t > 0 i =1 Histogram of s ( x ): r ( s ) = ( r 1 r 2 ... r t ) Define q ρ ( s ) = i ∈ Z + : d i =1 r i = max max i =1 r i . Example: Berman, Roth (Technion) ISIT 2020 14 / 36
s ( x ) = x 3 · ( x +1) 2 · ( x 2 + x +1) · ( x 3 + x 2 +1) · ( x 4 + x +1) 5 r i r ( s ) = (3 , 2 , 1 , 0 , 1 , 1) ρ ( s ) = 3 4 3 2 1 0 d i : 1 1 2 3 3 4 4 4 5 P i [ x ] : x x 3 + x 2 +1 x +1 x 2 + x +1 x 3 + x +1 x 4 + x +1 Berman, Roth (Technion) ISIT 2020 15 / 36
Maximal polynomial for m = 180 r i 10 9 8 7 6 5 4 3 2 1 0 d i : 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 Berman, Roth (Technion) ISIT 2020 16 / 36
Explicit formula for Φ( s ) It is easy to see that t ∏ Φ( s ) = ( r i +1) i =1 Berman, Roth (Technion) ISIT 2020 17 / 36
Explicit formula for Φ( s ) It is easy to see that t ∏ Φ( s ) = ( r i +1) i =1 = ⇒ deg s = m . Berman, Roth (Technion) ISIT 2020 17 / 36
Structural properties of maximal polynomials Histogram is monotonically decreasing . Histogram is all-positive . Berman, Roth (Technion) ISIT 2020 18 / 36
Structural properties of maximal polynomials Histogram is monotonically decreasing . Histogram is all-positive . General proof method Assume s ∈ P m does not satisfy the property. Construct from s a polynomial ˜ s ∈ P m for which Φ(˜ s ) > Φ( s ). Conclude that s cannot be maximal. Berman, Roth (Technion) ISIT 2020 18 / 36
Structural properties of maximal polynomials (continued) Degree vs. multiplicity of an irreducible factor For every i ∈ [1 : t ]: r i +2 ≤ d i < ρ +1 ρ +1 . r i Equivalently: r i ∈ {⌊ ρ / d i ⌋ , ⌊ ρ / d i ⌋− 1 } . Maximal degree of irreducible factor � � � � log q ( m / 8) < d t ≤ d t +1 ≤ log q m +1 . Estimation of ρ ρ = log q m � � ± O log q log q m . ln2 Berman, Roth (Technion) ISIT 2020 19 / 36
Maximal polynomial for m = 180 r i 10 9 log 2 (180) = 7 . 491 d t = 7 8 7 log 2 (180) / ln2 = 10 . 808 ρ = 9 6 5 4 3 2 1 0 d i : 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 Berman, Roth (Technion) ISIT 2020 20 / 36
Improved characterization for large d i Theorem � � For every i ∈ [1 : t ] such that d i ≥ Θ log q log q m : � � 1 � � log 2 1+ · log q m − O (1) r i +1 � 1+ 1 � � � < d i ≤ log 2 · log q m + O (1) . r i � �� 2 ( d i ± O (1)) / ⌊ log q m ⌋ − 1 �� Equivalently: r i = 1 . Berman, Roth (Technion) ISIT 2020 21 / 36
“Long tail” of 1-s r i > 1 only when d i log q m < log 2 (3 / 2)+ o m (1) ≈ 0 . 585 . Recall d t ≈ log q 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
Characterization of n -Maximal Polynomials Berman, Roth (Technion) ISIT 2020 23 / 36
Reminder: n -maximal polynomial Given n ∈ Z + and a polynomial s ( x ) ∈ P 2 n (i.e. deg s ≤ 2 n ): ( n , n )-factorization: An ( n , n ) -factorization of s ( x ) is an ordered pair ( u ( x ) , v ( x )) ∈ P n × P n s.t. s ( x ) = u ( x ) · v ( x ) . s ( x ) ∈ P 2 n is n -maximal if s ( x ) has maximal # of ( n , n )-factorizations. Berman, Roth (Technion) ISIT 2020 24 / 36
Notation For n ∈ Z + and s ( x ) = ∏ t i =1 p i ( x ) r i ∈ P 2 n : r 0 = 2 n − deg s d 0 ≡ 1 r n ( s ) = ( r 0 r ( s )) = ( r i ) t i =0 ρ n ( s ) = max { r 0 , ρ ( s ) } = max i ∈ Z ≥ 0 : d i =1 r i Berman, Roth (Technion) ISIT 2020 25 / 36
s ( x ) = x · ( x +1) 3 · ( x 2 + x +1) 2 · ( x 3 + x +1) n = 7 , 2 n = 14 , deg s = 11 4 r i r ( s ) = (1 , 3 , 2 , 1) 3 2 1 0 1 1 2 3 3 4 4 4 d i : P i [ x ] : x x 3 + x 2 +1 x +1 x 2 + x +1 x 3 + x +1 x 4 + x +1 Berman, Roth (Technion) ISIT 2020 26 / 36
s ( x ) = x · ( x +1) 3 · ( x 2 + x +1) 2 · ( x 3 + x +1) n = 7 , 2 n = 14 , deg s = 11 , r 0 = 3 4 r i r n ( s ) = (3 , 1 , 3 , 2 , 1) ρ n ( s ) = 3 3 2 1 0 1 1 1 2 3 3 4 4 d i : P i [ x ] : y x x 3 + x 2 +1 x +1 x 2 + x +1 x 3 + x +1 x 4 + x +1 Berman, Roth (Technion) ISIT 2020 27 / 36
Results Same structural properties as for maximal polynomials, where we replace: m ← → 2 n r ( s ) ← → r n ( s ) ρ ( s ) ← → ρ n ( s ) 1 ≤ i ≤ t ← → 0 ≤ i ≤ t . Berman, Roth (Technion) ISIT 2020 28 / 36
Results Same structural properties as for maximal polynomials, where we replace: m ← → 2 n r ( s ) ← → r n ( s ) ρ ( s ) ← → ρ n ( s ) 1 ≤ i ≤ t ← → 0 ≤ i ≤ t . Ordinary factorizations: t ∏ Φ( s ) = ( r i +1) . i =1 ( n , n ) -factorizations: Φ n ( s ) =??? Therefore, here we need more intricate proofs . Berman, Roth (Technion) ISIT 2020 28 / 36
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