LCD codes over F q are as good as linear codes for q at least four - - PowerPoint PPT Presentation
LCD codes over F q are as good as linear codes for q at least four Ruud Pellikaan g.r.pellikaan@tue.nl International Conference on Graph Theory and Information Security ICGTIS, August 7, 2017 Universitas Indonesia, Depok, Indonesia Faculteit
Faculteit Wiskunde & Informatica
Ruud Pellikaan g.r.pellikaan@tue.nl International Conference on Graph Theory and Information Security ICGTIS, August 7, 2017 Universitas Indonesia, Depok, Indonesia
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Communication: internet, telephone, WiFi, computer Memory: computer, compact disc, DVD, USB stick Barcodes, ISBN, product codes, QR codes ...
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source encoding sender noise receiver decoding target
✲
message
✲
001...
✲
011...
✲
message
✻
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Q alphabet of q elements Hamming distance d(x, y) = |{i | xi = yi}| between x = (x1, . . . , xn) and y = (y1, . . . , yn) in Q n
x
d(x,y)
❍❍❍❍❍ ❍ ❥
y
❍ ❍ ❍ ❍ ❍ ❍ ❨
d(y,z)
✘✘✘✘✘✘✘✘✘✘✘ ✘ ✿ z ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✾
d(x,z)
Triangle inequality
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C is called (block) code if it is a subset of Q n The minimum distance of C is: d(C) = min { d(x, y) | x, y ∈ C, x = y } parameters of C are (n, M, d)q or (n, M, d) q = |Q| = size of alphabet Q n = length of C M = |C| = size of C d = d(C) = minimum distance of C
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Fq the finite field with q = pe elements and p prime Fn
q is an Fq-linear vector space of dimension n
A linear code is an Fq-linear subspace C of Fn
q
with parameters [n, k, d]q or [n, k, d] or [n, k] q = size finite field n = length of C k = dimension of C d = minimum distance of C r = redundancy of C = n − k
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Let C be an [n, k] linear code over Fq Then G is a generator matrix of C if it is a k × n matrix with entries in Fq such that its rows are a basis of C Let m = (m1, . . . , mk) ∈ Fk
q be a message
Then c = mG is a codeword Fk
q −
→ Fn
q
Encoding m → mG = c So C is the image of Fk
q under G
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Let C be an [n, k] linear code over Fq Then H is called a parity check matrix of C if it is a (n − k) × n matrix with entries in Fq such that r ∈ C if and only if rH T = 0 Fn
q −
→ Fn−k
q
r → rH T r ∈ C if and only if syndrome cH T is zero So C is left kernel (null space) of H T
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The standard inner product is defined by a · b = a1b1 + · · · + anbn Is bilinear and non-degenerate but positive definite makes no sense not right picture Vectors a, b ∈ Fn
q are perpendicular denoted by a ⊥ b
if and only if a · b = 0
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Let C be a linear code in Fn
q
The dual code is defined by C ⊥ = { x ∈ Fn
q | x · c = 0 for all c ∈ C }
PROPOSITION Let C be an [n, k] code with generator matrix G Then C ⊥ is an [n, n − k] code with G as parity check matrix
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The code C is called linear with complementary dual (LCD) if C ∩ C ⊥ = {0} PROPOSITION (1992 Massey) LCD codes are asymptotically good (2004 Sendrier) LCD codes meet the Gilbert-Varshamov bound
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The hull of an Fq-linear code C is defined by H(C) = C ∩ C ⊥ Hence C is LCD if and only if H(C) = {0}
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PROPOSITION Let C be an Fq-linear [n, k] code Let h be the dimension of H(C) and r = k − h Then C has a generator matrix G0 such that G0G T
0 =
Oh×h Oh×r Or×h P
where Ol×m is the all zeros l × m matrix and P is an invertible r × r matrix Furthermore the rank of G1G T
1 is r for every generator matrix G1 of C
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COROLLARY Let C be an Fq-linear [n, k] code with generator matrix G Then the following statements are equivalent:
◮ C is LCD ◮ C ∩ C ⊥ = {0} ◮ GG T has rank k ◮ GG T is invertible
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Let C be the binary [7, 4, 3] Hamming code with generator matrix G1 = 1 1 1 1 1 1 1 1 1 1 1 1 1 Then G1G T
1 =
1 1 1 1 1 1 1 1 1 1 1 1 has rank 1 Hence H(C) has dimension 3
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Now C has another generator matrix G0 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 with G0G T
0 =
1
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◮ A permutation matrix is a square matrix with zeros and ones
such that in every row (and in every column) there is exactly one element equal to one
◮ A diagonal matrix is a square matrix with zeros outside its diagonal ◮ A monomial matrix is a square matrix
such that in every row (and in every column) there is exactly one nonzero element A permutation matrix and an invertible diagonal matrix are special monomial matrices
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Let C1 and C2 be Fq-linear codes of length n Then C1 and C2 are called
◮ permutational equivalent
if there exists a permutation matrix P such that C1P = C2
◮ diagonal equivalent
if there exists an invertible diagonal matrix D such that C1D = C2
◮ linear equivalent or monomial equivalent
if there exists a monomial matrix M such that C1M = C2
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The dimension of the hull of a code is
◮ invariant under permutational equivalence ◮ also invariant under monomial equivalence if q = 2, 3 ◮ can be computed with the (extended) weight enumerator ◮ is used to find the permutation in case
C1 and C2 are permutational equivalent
◮ is not a monomial equivalence invariant if q ≥ 4
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Let x, y ∈ F2 Define x ⊕ y ∈ Z by x y x ⊕ y 1 1 1 1 1 1 2 Let x, y ∈ Fn
2
Define x ⊕ y = (x1 ⊕ y1, . . . , xn ⊕ yn)
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Let C and D be Fq-linear codes of length n Define C × D = { (c, d) | c ∈ C, d ∈ D } C ⊕ D = { c ⊕ d | c ∈ C, d ∈ D } C ⊕ D is called unique decodable if the map C × D → C ⊕ D given by (c, d) → c ⊕ d is injective C ⊕ D is unique decodable if and only if C ∩ D = {0} Hence C ⊕ C ⊥ is unique decodable if and only if C is LCD
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Carlet and Guilley (2014) Let C and D be Fq-linear codes of length n Define C + D = { c + d | c ∈ C, d ∈ D } If C ∩ D = {0} then C + D is denoted by C ⊎ D Then C ⊎ D = Fn
q if and only if C ∩ D = {0} and dim C + dim D = n
Hence C ⊎ C ⊥ = Fn
q if and only if C is LCD
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Let x, y ∈ Fn
q
Then the star product is defined by x ∗ y = (x1y1, . . . , xnyn) Let x ∈ Fn
q have nonzero entries
Define x−1 = (x−1
1 , . . . , x−1 n )
Let C ⊆ Fn
q
Define x ∗ C = { x ∗ c | c ∈ C } C1 and C2 are scalar equivalent if and only if there exists an x with nonzero entries such that C2 = x ∗ C1
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THEOREM (2017 Carlet-Mesnager-Tang-Qi-P) If q ≥ 4 and C is an Fq-linear code Then there exits an x ∈ Fn
q with nonzero entries
such that x ∗ C is an LCD code Hence LCD codes over Fq are as good as Fq-linear codes if q ≥ 4
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THEOREM (2017 Carlet-Mesnager-Tang-Qi-P) If q ≥ 4 and C is an Fq-linear code Then there eixts an x ∈ Fn
q with nonzero entries
such that x ∗ C is an LCD code Hence LCD codes over Fq are as good as Fq-linear codes if q ≥ 4
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PROPOSITION Let f(X) be a nonzero polynomial of Fq[X1, . . . , Xn] such that the degree of f(X) with respect to Xj is at most q − 1 for all j Then there exists a x ∈ Fn
q such that f(x) = 0
PROPOSITION Let f(X) be a nonzero polynomial of Fq[X1, . . . , Xn] such that the degree of f(X) with respect to Xj is at most q − 2 for all j Then there exists a x ∈ Fn
q with nonzero entries such that f(x) = 0
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We may assume that C has a generator matrix of the form G = (Ik|B) Let x = (x1, . . . , xk) be an k-tuple of nonzero elements of Fq Let D(x) be the diagonal matrix with x on its diagonal Let Gx = (D(x)|B) be the generator matrix of the code Cx Then Cx is monomial equivalent with C Now det(GxG T
x ) = det(D(x2 1, . . . , x2 k ) + BB T)
is a polynomial in x1, . . . , xk Its degree with respect to xi is 2 for all i which is at most q − 2, since q ≥ 4 Hence there exists a x ∈ Fn
q with nonzero entries
such that det GxG T
x = 0
So GxG T
x is invertible
Therefore Cx is LCD
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