On functional codes and quantum codes

INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

On functional codes and quantum codes Daniele Bartoli Universit`a degli Studi di Perugia

GIORNATE DI GEOMETRIA COMBINATORIA VICENZA 13-14 FEBBRAIO 2012

Daniele BARTOLI

ON FUNCTIONAL CODES AND QUANTUM CODES

INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Outline 1

Introduction

2

Functional codes Preliminaries Ch (X )

3

Quantum codes Historical outline Spectrum Geometric constructions

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Introduction Definition A [n, k]q -linear code C is a k-dimensional subspace of GF (q)n .

Definition (Minimum distance) Let C be an [n, k]q -code. Let v, w ∈ C. The Hamming distance d(v, w) is defined as d(v, w) = #{i ∈ [1, . . . , n]|vi 6= wi }. The minimum distance of C is d = minv6=w∈C d(v, w). Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Remark An [n, k, d]q -code C can correct

Daniele BARTOLI

d−1 2

errors.


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Functional codes

K algebraically closed field of characteristic p > 0. X projective algebraic variety defined over Fq . Fq -rational points of X . Aut(X ) K-automorphism group of X .

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Example: Goppa codes

Fq (X ) field of rational functions of X defined over Fq DivFq (X ) group of Fq -divisors of X P1 , . .P . , Pn Fq -rational points of X and G = nP P ∈ DivFq (X ) with nPi = 0 for all i = 1, . . . , n. L(G ) = {f ∈ Fq (X ) \ 0 : G + (f ) ≥ 0} ∪ {0} e : L(G ) → Fnq f 7→ (f (P1 ), . . . , f (Pn )) k = `(G ) − `(G − D) and d ≥ n − deg (G )

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Functional codes of type Ch (X ) Definition Let X ⊂ PG (n, q) be a fixed algebraic variety and let {P1 , P2 , . . . , PN } its Fq -rational points. The functional code Ch (X ) is defined as {(f (P1 ), . . . , f (PN ))|f ∈ Fh } ∪ {0}, where Fh is the set of all the homogeneous polynomials of degree h over Fq in the variables X0 , . . . , Xn .

The code Ch (X ) has length N and dimension Daniele BARTOLI

n+h h

over Fq .


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Ch (X ): known results Code C2 (Qn ) C2 (Hn ) Ch (Hn ) Cherm (Hn ) Cherm (Qn )

Reference [EHRS2010 2] [HS2010] [ELX2011] [EHRS2010 1]

BBFS2011

[EHRS2010 1] F.A.B. Edoukou, A. Hallez, F. Rodier, and L. Storme, On the small weight codewords of the functional codes Cherm (X ), X a non-singular Hermitian variety, Des. Codes Cryptogr. 56 (2010), 219-233. [EHRS2010 2] F.A.B. Edoukou, A. Hallez, F. Rodier e L. Storme, A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2 (Q), Q a non-singular quadric, J. Pure Applied Algebra 214 (2010), 1729-1739. [HS2010] A. Hallez e L. Storme, Functional codes arising from quadric intersection with Hermitian varieties, Finite Fields Appl. 16 (2010), 27-35. [ELX2011] F.A.B. Edoukou, S. Ling, C. Xing, Structure of functional codes defined on non-degenerate Hermitian varieties, Journal of Combinatorial Theory, Series A 118 (2011), 2436-2444. [L1996] G. Lachaud, Number of points of plane sections and linear codes defined on algebraic varieties, Arithmetic, Geometry, and Coding Theory (Luminy, France, 1993), Walter De Gruyter, Berlin-New (1996), 77-104. DanieleYork BARTOLI ON FUNCTIONAL CODES AND QUANTUM CODES

INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Remark The weight of a codeword of Ch (X ) is equal to the number of points P1 , . . . , PN such that f (Pi ) 6= 0, i.e. N − |X ∩ Y|, where Y : f = 0. Then the minimum weight corresponds to varieties Y such that |X ∩ Y| is maximal. Idea |X ∩ Y| is maximal when Y splits in h hyperplanes. Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

CHerm (Qn ): results

n 2 3

4

≥5

H(n, q 2 ) Any Non-singular PH(2, q 2 ) LH(1, q 2 ) Non-singular PH(3, q 2 ) LH(2, q 2 ) πH(1, q 2 )

|Q ∩ H| ≤ 2(q + 1) ≤ 2q 3 + q 2 + 1 ≤ 2q 3 + 2 ≤ q 3 + 3q 2 − q + 1 ≤ q 5 + q 4 + 4q 3 − 3q + 1 ≤ q 5 + 2q 4 − 13 q + 2q 2 + q + 1 ≤ q5 + q3 + q2 + 1 ≤ q 5 + q 4 + q 3 + 2q 2 + 1

Non-singular

≤ q 2n−3 + q 2n−4 + 4q 2n−5 − 2q 2n−7 + 2q 2n−8 + q 2n−9 + . . . + q n−2 + 2q n−3

PH(n − 1, q 2 ) LH(2, q 2 ) πs H(n − s − 1, q 2 )

≤ q 2n−3 + q 2n−4 + 4q 2n−5 − 2q 2n−7 + 2q 2n−8 ≤ q 2n−3 + 2q 2n−5 + q 2n−6 + 2q 2n−8 + q 4 + q 2 + 1 ≤ q 2n−3 + 3q 2n−5 + 3q 2n−6 + 1

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

CHerm (Qn ): results

Theorem Let Q(n, q 2 ) be a non-singular quadric in PG (n, q 2 ) and H(n, q 2 ) an Hermitian variety in PG (n, q 2 ). Then q 2n−3 + q 2n−4 + 4q 2n−5 − 2q 2n−7 + |Q(n, q ) ∩ H| ≤ W n = . 2q 2n−8 + q 2n−9 + . . . + q n−2 + 2q n−3 2

The code CHerm (Q(n, q 2 )) has minimum distance at least |Q(n, q 2 )| − W n .

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Historical outline 1980s : Feynman’s ideas

1994 : Shor’s algorithm 1995 : Shor’s [[9, 1, 3]]-code 1997 : Calderbank, Rains, Shor, Sloane : translation from physics to mathematics

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Historical outline

1980s : Feynman’s ideas 1994 : Shor’s algorithm

148656260640061 = 45681901 ∗ 3254161 1995 : Shor’s [[9, 1, 3]]-code 1997 : Calderbank, Rains, Shor, Sloane : translation from physics to mathematics

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Historical outline 1980s : Feynman’s ideas 1994 : Shor’s algorithm 1995 : Shor’s [[9, 1, 3]]-code

1997 : Calderbank, Rains, Shor, Sloane : translation from physics to mathematics

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Historical outline 1980s : Feynman’s ideas 1994 : Shor’s algorithm 1995 : Shor’s [[9, 1, 3]]-code 1997 : Calderbank, Rains, Shor, Sloane : translation from physics to mathematics

=⇒

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Quantum codes

Definition A quaternary quantum stabilizer code is an additive quaternary code C contained in its dual C ⊥ , where the duality is with respect to the symplectic form. Definition An [[n, k, d]]-quantum code, with k > 0, has binary dimension n − k and every codeword of C ⊥ having weight at most d − 1 belongs to C. The code is said pure if C ⊥ \ {0} does not contain codewords of weight less than d. An [[n, 0, d]]-quantum code C is a code with C = C ⊥ of strength t = d − 1.

Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Theorem The following are equivalent 1

an [[n, k, d]]-quantum code pure;

2

an [n, k]4 -linear code with codewords of even weight;

3

a set P of n points in PG ( n−k 2 − 1, 4) of strenght t = d − 1 such that for each hyperplane H, |P ∩ H| has the same parity of n.

t

t t H t

t

t

t t t

t t

[BFGMP2008] J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, The geometry of quantum codes, Innov. Incidence Geom. 6/7 (2007/2008), 53-71. Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Link with caps

u u

u u H

u

u u

u u

u u

pure [[n, n − 10, 4]]-quantum codes m n-caps in PG (4, 4) satisfying (3) Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Results: spectrum Theorem Let K ⊂ PG (4, 4) be a quantum k-cap. Then k ∈ {10, 12 − 36, 38, 40, 41}. [BBMP2010] D. B., J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes, Error-Correcting Codes, Finite Geometries and Cryptography, AMS, Series: Contemporary Mathematics 523, Eds. Aiden A. Bruen and David L. Wehlau (2010), 149-154. [BFMP2011] D. B., G. Faina, S. Marcugini e F. Pambianco, New quantum caps in PG (4, 4), submitted. [BBEFMP2012] D. B., J. Bierbrauer, Y. Edel, G. Faina, S. Marcugini and F. Pambianco, The structure of quaternary quantum caps, preprint. Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Results: general constructions Theorem Let K1 , K2 two quantum sets of PG (m − 1, 4). Then K1 ∪ K2 and K2 \ K1 are quantum sets.

[BBMP2010] D. B., J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes, Error-Correcting Codes, Finite Geometries and Cryptography, AMS, Series: Contemporary Mathematics 523, Eds. Aiden A. Bruen and David L. Wehlau (2010), 149-154. Daniele BARTOLI


INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Results: general constructions Theorem Let H1 , H2 two different hyperplanes in PG (m, 4) and Ki ⊂ Hi two quantum caps such that K1 ∩ H1 ∩ H2 = K2 ∩ H1 ∩ H2 . Then their symmetric sum (K1 \ K2 ) ∪ (K2 \ K1 ) is a quantum cap.



INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Results: general constructions Theorem Let H1 , H2 two different hyperplanes in PG (m, 4), S = H1 ∩ H2 . Let K1 ⊂ H1 a quantum cap in H1 and K2 ⊂ H2 \ S an affine quantum cap. If K2 ∪ (K1 ∩ S) is a cap, then K1 ∪ K2 is a quantum cap.



INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Results: general constructions Theorem Let S1 , S2 due different secunda of PG (m, 4) in general position. Let Ki ⊂ Si two quantum caps such that K1 ∩ S1 ∩ S2 = K2 ∩ S1 ∩ S2 . Then their symmetric sum is a quantum cap.



INTRODUCTION

FUNCTIONAL CODES

QUANTUM CODES

Thank you!

Daniele BARTOLI
