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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 coding1,
• space–time codes2,
• cryptography3,
• low rank matrix recovery4
• Gabidulin codes5 (over finite fields) are the rank–metric analog of Reed–Solomon codes
• Recently, they were extended6 to fields of characteristic zero via field automorphisms.
• Independently, support constrained Reed–Solomon codes7 and Gabidulin codes8 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

1Silva et al. TIT’08 2Lusina et al. TIT’03 3Gabidulin et al. ’91 4M¨

uelich et al. ’17

5Delsarte ’78, Gabidulin ’85 6Augot et al. ’18 7Dau et al. ISIT’14, Halbawi et al. ISIT’14, Yildiz et al. TIT’18, Lovett FOCS’18 8Yildiz 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
• perations from E.
• E is a vector space over F.

x ∈ E − → (x1, . . . , xm) ∈ Fm

• 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 θ0, θ1, θ2, . . . , θm−1

Examples

• C/R,

dim = 2 θ(x) = x, θ(x) = x∗

• Fqm/Fq,

dim = m θ(x) = xqi, i = 0, 1, . . . , m − 1

• Q(e2π/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 = dimF E. Let C ⊂ En be a k–dimensional linear subspace. Codewords are in En or Fm×n dH = min

0=c∈C wt(c)

considering c as a vector in En dR = min

0=c∈C rank(c)

considering c as a matrix in Fm×n

Singleton Bound

dR ≤ dH ≤ n − k + 1 dH = n − k + 1 = ⇒ MDS (maximum distance separable), e.g. Reed–Solomon codes dR = 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 Gk×n Gk×n =      × × · · · × · · · × . . . . . . . . . . . . × × · · ·      → Z1 → Z2 . . . → Zk Zi: Set of zero locations in the ith row.

Objective

Given Z1, . . . , Zk ⊂ {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 fields10
• Over fields of characteristic zero

9Yildiz et al. TIT’18, Lovett FOCS’18 10Yildiz et al. TIT’19

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 6 / 17

Necessary Condition

Gk×n =      × × · · · × · · · × . . . . . . . . . . . . × × · · ·      → Z1 → Z2 . . . → Zk Zi: Set of zero locations in the ith row.

MDS condition [Dau et al. ISIT’14]

For every nonempty Ω ⊂ {1, 2, . . . , k},

• i∈Ω

Zi

• ≤ k − |Ω|

i.e.: Not too many zeros: |Zi| ≤ k − 1 (each row has at most k − 1 zeros) |Zi ∩ Zj| ≤ k − 2, i = j (each pair of rows has at most k − 2 common zeros) . . . |Z1 ∩ · · · ∩ Zk| = 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)        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αk−1

1

αk−1

2

· · · αk−1

n

              θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θk−1(α1) θk−1(α2) · · · θk−1(αn)        α1, . . . , αn ∈ F are distinct α1, . . . , αn ∈ E are linearly independent over F (For finite fields Fqm/Fq, θ(x) = xq)

• 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. G =      × × · · · × · · · × . . . . . . . . . . . . × × · · ·      = Tk×k ·      a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . . . . ak1 ak2 · · · akn      Assuming |Zi| = k − 1, Ti,:

• 1×k

· A:,Zi

k×(k−1)

= Gi,Zi = 0 Ti,j = (−1)j+1 det A−j,Zi 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 αi’s such that

• n × n extension of A is full rank
• T is full rank, where T = [(−1)j+1 det A−j,Zi]i,j=1,...,k

(i) Reed–Solomon codes

(ii) Gabidulin codes A(i) =        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αk−1

1

αk−1

2

· · · αk−1

n

       , A(ii) =        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θk−1(α1) θk−1(α2) · · · θk−1(αn)       

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 αi’s such that P(X) = det X · det T = det X · det[(−1)j+1 det X[k]−j,Zi]i,j∈[k] = 0

(i) Reed–Solomon codes

(ii) Gabidulin codes X(i) =        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αn−1

1

αn−1

2

· · · αn−1

n

       , X(ii) =        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θn−1(α1) θn−1(α2) · · · θn−1(αn)        P(X) is a multivariate polynomial in n2 variables defined by the zero pattern (i.e. the subsets Z1, . . . , Zk)

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 α1, . . . , αn such that P        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αn−1

1

αn−1

2

· · · αn−1

n

       = 0 if and only if

• i∈Ω

Zi

• ≤ k − |Ω|

∀∅ = Ω ⊂ [k]

• 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

Idea

Question

For the following values of X, do there exist αi’s such that P(X) = 0?

(i) Reed–Solomon codes

(ii) Gabidulin codes X(i) =        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αn−1

1

αn−1

2

· · · αn−1

n

       , X(ii) =        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θn−1(α1) θn−1(α2) · · · θn−1(αn)        ‘Yes’ to any of these questions = ⇒ P is not the zero polynomial

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 13 / 17

Idea

Question

For the following values of X, do there exist αi’s such that P(X) = 0?

(i) Reed–Solomon codes

(ii) Gabidulin codes X(i) =        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αn−1

1

αn−1

2

· · · αn−1

n

       , X(ii) =        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θn−1(α1) θn−1(α2) · · · θn−1(αn)        MDS cond.

GM-MDS Conj.

= = = = = = = = = ⇒ ‘Yes’ for RS codes = ⇒ P = 0 = ⇒ ‘Yes’ for Gabidulin in char. zero

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 13 / 17

Idea

Question

For the following values of X, do there exist αi’s such that P(X) = 0?

(i) Reed–Solomon codes

(ii) Gabidulin codes X(i) =        1 1 · · · 1 α1 α2 · · · αn α2

1

α2

2

· · · α2

n

. . . . . . . . . αn−1

1

αn−1

2

· · · αn−1

n

       , X(ii) =        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θn−1(α1) θn−1(α2) · · · θn−1(αn)        MDS cond.

GM-MDS Conj.

= = = = = = = = = ⇒ ‘Yes’ for RS codes = ⇒ P = 0 = ⇒ ‘Yes’ for Gabidulin in char. zero Claim: If P is nonzero, we can get P(X) = 0 w.h.p. by randomly choosing αi’s for Gabidulin codes in characteristic zero.

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 13 / 17

Schwartz–Zippel Lemma

Lemma

Let f be a nonzero polynomial over F in n variables. Let x1, . . . , xn be uniformly and independently chosen from a finite nonempty subset S ⊂ F. Then, P(f(x1, . . . , xn) = 0) ≤ deg f |S|

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 14 / 17

Schwartz–Zippel Lemma

Lemma

Let f be a nonzero polynomial over F in n variables. Let x1, . . . , xn be uniformly and independently chosen from a finite nonempty subset S ⊂ F. Then, P(f(x1, . . . , xn) = 0) ≤ deg f |S| Unfortunately, we cannot choose the entries of X independently, X must be in the following form:        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θn−1(α1) θn−1(α2) · · · θn−1(αn)       

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 14 / 17

Schwartz–Zippel Lemma for automorphisms

For a fixed basis for E, every number in E can be written as a vector with entries from F.

Lemma

Let f be a nonzero polynomial over E in mn variables. Let α1, . . . , αn ∈ E such that the entries of the αi’s in the vector form are uniformly and independently chosen from a finite nonempty subset S ⊂ F. Then, P        f        θ0(α1) θ0(α2) · · · θ0(αn) θ1(α1) θ1(α2) · · · θ1(αn) θ2(α1) θ2(α2) · · · θ2(αn) . . . . . . . . . θm−1(α1) θm−1(α2) · · · θm−1(αn)        = 0        ≤ deg f |S| (1)

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 15 / 17

Code Construction

Inputs: A finite nonempty set S ⊂ F and subsets Z1, . . . , Zk ⊂ [n] satisfying the MDS condition. Steps:

• Choose (γij)i∈[n],j∈[m] uniformly and independently at random from S.
• Let αi = m

j=1 γijbj for i ∈ [n] for a basis {b1, . . . , bm} for E.

• Construct A ∈ Ek×n as the standard generator matrix of the Gabidulin code with

parameters α1, . . . , αn.

• Add elements to the Zi’s from [n] (if necessary) by following the algorithm given by Dau

et al. so that |Zi| = k − 1 and they still satisfy the MDS condition.

• Define T ∈ Ek×k as

Tij = (−1)j+1 det A−j,Zi, i, j ∈ [k] (2) Output: The generator matrix G = T · A ∈ Ek×n.

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 16 / 17

Conclusion

• MDS condition is necessary and sufficient also for the existence of Gabidulin codes in

characteristic zero

• Randomized construction algorithm whose probability of success is arbitrarily close to 1
• This was not the case for the Reed–Solomon codes or Gabidulin codes in finite fields.
• We take the advantage of infinite field size!

Yildiz-Raviv-Hassibi Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero ISIT ’20 17 / 17

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