Support Constrained Generator Matrices of Gabidulin Codes in - PowerPoint PPT Presentation

Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero Hikmet Yildiz ∗ , Netanel Raviv † , Babak Hassibi ∗ ∗ California Institute of Technology, Pasadena CA † Washington University in Saint Louis, St. Louis MO ISIT, Los Angeles CA June 2020 Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 1 / 17

Outline • Introduction and Motivation • Preliminaries • Field extensions • Rank–metric codes • Problem Definition and Results • Related problems on support constrained generator matrices • Known results on Reed–Solomon codes and Gabidulin codes in finite fields • Our case: Gabidulin codes in characteristic zero • Conclusion Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 2 / 17

Introduction and Motivation • Rank–metric codes have applications in • network coding 1 , • space–time codes 2 , • cryptography 3 , • low rank matrix recovery 4 • Gabidulin codes 5 (over finite fields) are the rank–metric analog of Reed–Solomon codes • Recently, they were extended 6 to fields of characteristic zero via field automorphisms. • Independently, support constrained Reed–Solomon codes 7 and Gabidulin codes 8 are studied lately due to their applications in distributed computing. • We intend to extend these results on Gabidulin codes to the fields of characteristic zero 1 Silva et al. TIT’08 2 Lusina et al. TIT’03 3 Gabidulin et al. ’91 4 M¨ uelich et al. ’17 5 Delsarte ’78, Gabidulin ’85 6 Augot et al. ’18 7 Dau et al. ISIT’14, Halbawi et al. ISIT’14, Yildiz et al. TIT’18, Lovett FOCS’18 8 Yildiz et al. TIT’19 Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 3 / 17

Field extensions • A field extension E of a field F (E / F) is a field such that F ⊂ E and F inherits the operations from E. • E is a vector space over F. → ( x 1 , . . . , x m ) ∈ F m x ∈ E − • An automorphism θ : E → E of E / F: θ ( x ) = x ∀ x ∈ F θ ( x + y ) = θ ( x ) + θ ( y ) ∀ x, y ∈ E θ ( xy ) = θ ( x ) θ ( y ) ∀ x, y ∈ E • Cyclic extension: all automorphisms are θ 0 , θ 1 , θ 2 , . . . , θ m − 1 Examples • C / R , θ ( x ) = x, θ ( x ) = x ∗ dim = 2 θ ( x ) = x q i , • F q m / F q , dim = m i = 0 , 1 , . . . , m − 1 • Q ( e 2 π/N ) / Q , dim = ϕ ( N ) Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 4 / 17

Rank–metric codes Let E / F and m = dim F E. Let C ⊂ E n be a k –dimensional linear subspace. Codewords are in E n or F m × n considering c as a vector in E n d H = min 0 � = c ∈C wt( c ) considering c as a matrix in F m × n d R = min 0 � = c ∈C rank( c ) Singleton Bound d R ≤ d H ≤ n − k + 1 d H = n − k + 1 = ⇒ MDS (maximum distance separable), e.g. Reed–Solomon codes d R = n − k + 1 = ⇒ MRD (maximum rank distance), e.g. Gabidulin codes MRD codes are MDS! Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 5 / 17

Problem Definition C = rowsp G k × n   × × · · · → Z 1 0 0 × · · · × → Z 2 0 0     G k × n = . . . . .  . . . .  . . . . . .   × × 0 · · · 0 → Z k Z i : Set of zero locations in the i th row. Objective Given Z 1 , . . . , Z k ⊂ { 1 , 2 , . . . , n } , • Complete G such that it is an MDS code (GM–MDS conjecture) 9 • Complete G such that it is an MRD code. • Over finite fields 10 • Over fields of characteristic zero 9 Yildiz et al. TIT’18, Lovett FOCS’18 10 Yildiz et al. TIT’19 Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 6 / 17

Necessary Condition   × 0 × · · · 0 → Z 1 0 0 × · · · × → Z 2     G k × n = . . . . .   . . . . . . . . . .   → Z k × × 0 · · · 0 Z i : Set of zero locations in the i th row. MDS condition [Dau et al. ISIT’14] For every nonempty Ω ⊂ { 1 , 2 , . . . , k } , � � � � � Z i � ≤ k − | Ω | � � � � � i ∈ Ω i.e. : Not too many zeros: |Z i | ≤ k − 1 (each row has at most k − 1 zeros) |Z i ∩ Z j | ≤ k − 2 , i � = j (each pair of rows has at most k − 2 common zeros) . . . |Z 1 ∩ · · · ∩ Z k | = 0 (there is no all zero column) Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 7 / 17

Reed–Solomon and Gabidulin codes Reed–Solomon codes (MDS) Gabidulin codes (MRD)     θ 0 ( α 1 ) θ 0 ( α 2 ) θ 0 ( α n ) 1 1 · · · 1 · · · · · · θ 1 ( α 1 ) θ 1 ( α 2 ) θ 1 ( α n ) α 1 α 2 α n · · ·         α 2 α 2 α 2 · · · θ 2 ( α 1 ) θ 2 ( α 2 ) θ 2 ( α n ) · · ·     1 2 n     . . . . . .  . . .   . . .  . . . . . .     α k − 1 α k − 1 α k − 1 θ k − 1 ( α 1 ) θ k − 1 ( α 2 ) θ k − 1 ( α n ) · · · · · · 1 2 n α 1 , . . . , α n ∈ F are distinct α 1 , . . . , α n ∈ E are linearly independent over F θ ( x ) = x q ) (For finite fields F q m / F q , • n × n extensions of the matrices above must be full rank. • We can multiply them by any k × k invertible transformation matrix from left. Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 8 / 17

Finding a transformation matrix Find a matrix T for a given matrix A such that G = T · A satisfies the given zero pattern.     × × · · · · · · 0 0 a 11 a 12 a 1 n × · · · × · · · 0 0 a 21 a 22 a 2 n         G =  = T k × k · . . . . . . .  . . . .   . . .  . . . . . . .    × × 0 · · · 0 a k 1 a k 2 · · · a kn Assuming |Z i | = k − 1 , T i, : · A : , Z i = G i, Z i = 0 ���� � �� � 1 × k k × ( k − 1) T i,j = ( − 1) j +1 det A − j, Z i i.e. T can be constructed from the ( k − 1) × ( k − 1) minors of A . Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 9 / 17

Question For the following values of A , do there exist α i ’s such that • n × n extension of A is full rank • T is full rank, where T = [( − 1) j +1 det A − j, Z i ] i,j =1 ,...,k (i) Reed–Solomon codes (ii) Gabidulin codes     · · · θ 0 ( α 1 ) θ 0 ( α 2 ) θ 0 ( α n ) 1 1 1 · · · θ 1 ( α 1 ) θ 1 ( α 2 ) θ 1 ( α n ) · · · α 1 α 2 α n · · ·         α 2 α 2 α 2 θ 2 ( α 1 ) θ 2 ( α 2 ) θ 2 ( α n ) · · · A (i) = , A (ii) = · · ·     1 2 n     . . . . . .  . . .   . . .  . . . . . .     α k − 1 α k − 1 α k − 1 θ k − 1 ( α 1 ) θ k − 1 ( α 2 ) θ k − 1 ( α n ) · · · · · · 1 2 n Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 10 / 17

Question For the following values of X , do there exist α i ’s such that P ( X ) = det X · det T = det X · det[( − 1) j +1 det X [ k ] − j, Z i ] i,j ∈ [ k ] � = 0 (ii) Gabidulin codes (i) Reed–Solomon codes     θ 0 ( α 1 ) θ 0 ( α 2 ) θ 0 ( α n ) 1 1 · · · 1 · · · θ 1 ( α 1 ) θ 1 ( α 2 ) θ 1 ( α n ) α 1 α 2 · · · α n · · ·         α 2 α 2 α 2 θ 2 ( α 1 ) θ 2 ( α 2 ) θ 2 ( α n ) X (i) = · · · , X (ii) = · · ·     1 2 n     . . . . . .  . . .    . . . . . . . . .     α n − 1 α n − 1 α n − 1 θ n − 1 ( α 1 ) θ n − 1 ( α 2 ) θ n − 1 ( α n ) · · · · · · 1 2 n P ( X ) is a multivariate polynomial in n 2 variables defined by the zero pattern (i.e. the subsets Z 1 , . . . , Z k ) Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 11 / 17

Known results • The answer is ‘Yes’ for Reed–Solomon codes iff the MDS condition holds, i.e. GM–MDS Conjecture [Yildiz et al. TIT’18, Lovett FOCS’18] There exists α 1 , . . . , α n such that   1 1 · · · 1 α 1 α 2 · · · α n     α 2 α 2 α 2 · · ·   P � = 0 1 2 n   . . .   . . . . . .   α n − 1 α n − 1 α n − 1 · · · 1 2 n if and only if � � � � � Z i � ≤ k − | Ω | ∀∅ � = Ω ⊂ [ k ] � � � i ∈ Ω • The answer is the same for Gabidulin codes over finite fields. [Yildiz et al. TIT’19] Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 12 / 17