restart Macaulay2, version 1.10 with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone ---Here are examples of quantum codes produced based on the work ---done in the paper: "Quantum Stabilizer and Subsystem codes from Algebro-geometric Toric codes" ---written jointly with Ravindra Girivaru. --- In these examples both values obtained as "Min Distance of Dual Code" and "Finer Min Distance of Dual --- Code" are lower bounds for the Minimum distance of the quantum code, with the latter being a finer bound. ---Examples of codes produced from the 2 dimensional projective space---- I. i2 : mindualdistP2sym(c, 5/4, 1/3) ell = 14 9 --c - - 15 5 ino= 11 11 - --c - -- 60 5 ino2= 3 -c - 4 4 The min dual distance >= 1 2 43 36 --c + --c + -- 60 60 5 Finer min dual distance >= 71 2 683 99 ---c + ---c + -- 900 300 25 The min height = 4 4 --c - -- 15 15 Dim of the Class code = 1 2 3 2 -c - -c - -- 5 5 15 Dim of the Quantum subsystem code = 12 2 24 16 --c + --c + -- 25 25 75 Dim of the Quantum stabilizer code = 3 2 6 4 -c + -c + -- 5 5 15 2 2 4 2 1 2 1 4 1 2 3 2 12 2 24 16 3 2 6 4 1 2 43 36 71 2 683 99 (--c - --c + --, , --c - -c - --, , -c - -c - --, , --c + --c + --, , -c + -c + --, , --c + --c + --, , ---c + ---c + --) 15 15 15 15 3 15 5 5 15 25 25 75 5 5 15 60 60 5 900 300 25 mindualdistP2(26, 5/4, 1/3)---with c=26 = 3^3-1 ell = 337 --- 15 ino= 209 - --- 30 ino2= 31 -- 2 The min dual distance >= 371 --- 10 Finer min dual distance >= 371 --- 10 The min height = 20 -- 3 Dim of the Class code = 1792 ---- 15 Dim of the Quantum subsystem code = 26224 ----- 75 Dim of the Quantum stabilizer code = 6556 ---- 15 (83.4, , 36.1, , 120, , 350, , 437, , 37.1, , 37.1) II. mindualdistP2(c, 12/10, 1/3) ell = 17 11 --c - -- 18 6 ino= 13 13 - --c - -- 90 6 ino2= 4 -c - 4 5 The min dual distance >= 1 2 53 22 --c + --c + -- 90 90 3 Finer min dual distance >= 103 2 599 143 ----c + ---c + --- 1620 270 36 The min height = 5 5 --c - -- 18 18 Dim of the Class code = 7 2 11 5 --c - --c - -- 36 18 36 Dim of the Quantum subsystem code = 22 2 44 2 --c + --c + - 45 45 9 Dim of the Quantum stabilizer code = 11 2 11 5 --c + --c + -- 18 9 18 5 2 5 5 1 2 1 5 7 2 11 5 22 2 44 2 11 2 11 5 1 2 53 22 103 2 599 143 (--c - --c + --, , --c - -c - --, , --c - --c - --, , --c + --c + -, , --c + --c + --, , --c + --c + --, , ----c + ---c + ---) 36 18 36 18 3 18 36 18 36 45 45 9 18 9 18 90 90 3 1620 270 36 mindualdistP2(15, 12/10, 1/3)---- with c=15 = 2^4-1 ell = 37 -- 3 ino= 13 - -- 3 ino2= 8 The min dual distance >= 56 -- 3 Finer min dual distance >= 56 -- 3 The min height = 35 -- 9 Dim of the Class code = 310 --- 9 Dim of the Quantum subsystem code = 1124 ---- 9 Dim of the Quantum stabilizer code = 1405 ---- 9 245 65 310 1124 1405 56 56 (---, , --, , ---, , ----, , ----, , --, , --) 9 9 9 9 9 3 3 ---Examples of codes produced from the n-dimensional projective spaces---- I. mindualdistPn(c, 5/2, 1/3, 3) --- i.e. Here n=3 ell = 14 2 23 --c - -- 15 15 The intersecton no = 1 -c - 5 2 The min dual distance >= 1 3 1 2 23 23 --c + -c + --c + -- 30 3 30 3 The finer min dual distance >= 43 3 52 2 989 1196 ---c + ---c + ---c + ---- 450 225 450 225 The height of the box= 1 4 --c + -- 15 15 The volume of the top pyramid = 4 3 4 2 4 4 --c - --c + --c - -- 45 15 15 45 The volume of the bottom box = 1 3 2 2 7 4 --c + --c - --c + -- 15 15 15 15 Dim of the Class code = 7 3 2 2 1 8 --c - --c - -c + -- 45 15 5 45 Dim of the Quantum subsystem code = 124 3 16 2 8 64 ---c + --c + --c - --- 225 75 25 225 Dim of the Quantum stabilizer code = 31 3 4 2 2 16 --c + --c + -c - -- 45 15 5 45 i3 : mindualdistPn(26, 5/2, 1/3, 3) --- with c=26= 3^3-1 ell = 3147 ---- 5 The intersecton no = 8 The min dual distance >= 4194 ---- = 839 5 The finer min dual distance >= 142363 ------ = 1900 75 The height of the box= 2 The volume of the top pyramid = 12500 ----- 9 The volume of the bottom box = 1250 Dim of the Class code = 23750 ----- = 2640 9 Dim of the Quantum subsystem code = 442736 ------ 45 Dim of the Quantum stabilizer code = 110684 ------ 9 II. mindualdistPnsym(c, 7/2, 1/3, 4) --- This is for the projective space of dimension 4 ell = 20 3 31 --c - -- 21 21 The intersecton no = 1 -c - 6 2 The min dual distance >= 1 4 2 3 31 62 --c + -c + --c + -- 42 7 42 7 The finer min dual distance >= 61 4 95 3 1891 2945 ---c + ---c + ----c + ---- 882 441 882 441 The height of the box= 1 2 --c + - 21 7 The volume of the top pyramid = 1 4 2 3 3 2 2 1 --c - -c + -c - -c + -- 14 7 7 7 14 The volume of the bottom box = 1 4 1 3 5 2 17 2 --c + -c - -c + --c - - 21 7 7 21 7 Dim of the Class code = 5 4 1 3 2 2 11 3 --c - -c - -c + --c - -- 42 7 7 21 14 Dim of the Quantum subsystem code = 64 4 8 3 16 2 88 12 ---c + --c + --c - ---c + -- 105 35 35 105 35 Dim of the Quantum stabilizer code = 16 4 2 3 4 2 22 3 --c + -c + -c - --c + - 21 7 7 21 7 mindualdistPn(26, 7/2, 1/3, 4) -with c=26=3^3-1 ell = 117163 ------ 7 The intersecton no = 7 The min dual distance >= 111511 ------ =15900 7 The finer min dual distance >= 1737224 ------- = 35500 49 The height of the box= 32 -- 21 The volume of the top pyramid = 390625 ------ 14 The volume of the bottom box = 500000 ------ 21 Dim of the Class code = 2171875 ------- = 51700 42 Dim of the Quantum subsystem code = 29698484 -------- 105 Dim of the Quantum stabilizer code = 7424621 ------- 21 ---Examples of codes produced from the 2 dimensional projective space with a point blown up---- I. mindualdistBP2(c, 5/4, 1/3) ell= 3 -c - 6 4 intersection no = 1 - --c + 5 12 The min height = 1 --c - 1 12 Dim of the Class code = 95 2 1 1 ---c - -c - - 288 4 2 Dim of the Quantum subsystem code = 49 2 2 4 ---c + -c + - 180 5 5 Dim of the Quantum stabilizer code = 49 2 1 ---c + -c + 1 144 2 Min Distance of Dual code = 1 2 3 --c - -c + 8 12 4 Finer Min distance of Dual code = 13 2 9 --c + -c - 28 48 4 1 2 3 23 2 1 95 2 1 1 49 2 2 4 49 2 1 1 2 3 13 2 9 (-c + -c - 1, , ---c - c + -, , ---c - -c - -, , ---c + -c + -, , ---c + -c + 1, , --c - -c + 8, , --c + -c - 28) 4 4 288 2 288 4 2 180 5 5 144 2 12 4 48 4 i5 : mindualdistBP2(26, 5/4, 1/3)--- with c=26=3^3-1 ell= 27 -- 2 intersection no = 17 -- 6 The min height = 7 - 6 Dim of the Class code = 15551 ----- 72 Dim of the Quantum subsystem code = 1757 ---- 9 Dim of the Quantum stabilizer code = 8785 ---- 36 Min Distance of Dual code = 269 --- 6 Finer Min distance of Dual code = 2563 ---- 12 375 2051 15551 1757 8785 269 2563 (---, , ----, , -----, , ----, , ----, , ---, , ----) 2 72 72 9 36 6 12 ---Note: In the last example, 2k= 432 and 2d= 428 so that this is an almost Quantum MDS code. II. mindualdistBP2sym(c, 6/5, 1/3) ell= 4 -c - 6 5 intersection no = 2 - --c + 5 15 The min height = 2 --c - 1 15 Dim of the Class code = 73 2 1 1 ---c - -c - - 225 5 2 Dim of the Quantum subsystem code = 316 2 8 4 ----c + --c + - 1125 25 5 Dim of the Quantum stabilizer code = 79 2 2 ---c + -c + 1 225 5 Min Distance of Dual code = 1 2 4 --c - -c + 8 15 5 Finer Min distance of Dual code = 17 2 14 --c + --c - 28 75 5 1 2 4 28 2 1 73 2 1 1 316 2 8 4 79 2 2 1 2 4 17 2 14 (-c + -c - 1, , ---c - c + -, , ---c - -c - -, , ----c + --c + -, , ---c + -c + 1, , --c - -c + 8, , --c + --c - 28) 5 5 225 2 225 5 2 1125 25 5 225 5 15 5 75 5 mindualdistBP2(15, 6/5, 1/3) with c=15=2^4-1 ell= 6 intersection no = 3 The min height = 1 Dim of the Class code = 139 --- 2 Dim of the Quantum subsystem code = 344 --- 5 Dim of the Quantum stabilizer code = 86 Min Distance of Dual code = 11 Finer Min distance of Dual code = 65 344 (56, , 13.5, , 69.5, , ---, , 86, , 11, , 65) 5 ---Note: In the last example, 2k= 139 and 2d= 130 so that this is also an almost Quantum MDS code. ---Examples of codes produced from the Hirzebruch surfaces. The last parameter denotes the type of the -- Hirzebruch surface, the firts examples being Hirzebruch surface F2. I. mindualdistFmsym(c, 4/5, 1/3, 2) ell= 2 -c - 4 3 ino = 1 --c 15 The min height = 1 1 --c - - 30 2 Dim of the Class code = 299 2 3 1 ---c - --c - - 900 10 4 Dim of the Quantum subsystem code = 8 2 8 4 --c - --c + - 25 25 5 Dim of the Quantum stabilizer code = 2 2 2 -c - -c + 1 5 5 Min distance of dual code = 14 2 136 --c + ---c - 32 45 15 mindualdistFm(1023, 4/5, 1/3, 2) -with c=1023= 2^10-1 ell= 678 ino = 341 --- 5 The min height = 168 --- 5 Dim of the Class code = 8684326 ------- = 347000 25 Dim of the Quantum subsystem code = 8364068 ------- 25 Dim of the Quantum stabilizer code = 2091017 ------- 5 Min distance of dual code = 334830 ---- Note: The last example is also one where the code is a near Quantum MDS code, since 2k= 694000 while 2d= 669660 II. mindualdistFm(c, 7/10, 1/3, 2) ell= 2 -c - 4 3 ino = 4 --c 15 The min height = 2 1 --c - - 15 2 Dim of the Class code = 71 2 1 1 ---c - -c - - 225 5 4 Dim of the Quantum subsystem code = 12 2 12 4 --c - --c + - 25 25 5 Dim of the Quantum stabilizer code = 3 2 3 -c - -c + 1 5 5 Min distance of dual code = 11 2 124 --c + ---c - 32 45 15 mindualdistFm(120, 7/10, 1/3, 2) -- with c=120= 11^2-1 ell= 76 ino = 32 The min height = 31 -- 2 Dim of the Class code = 18079 ----- 4 Dim of the Quantum subsystem code = 34276 ----- 5 Dim of the Quantum stabilizer code = 8569 Min distance of dual code = 4480 ---- Note: this is also a near Quantum MDS code since 2k= 9040 while 2d= 8960 III. mindualdistFmsym(c,11/20, 1/3, 3) --- This is a Hirzebruch surface of type F3. ell= 2 -c - 4 3 ino = 7 --c + 2 20 The min height = 7 1 --c - - 60 3 Dim of the Class code = 751 2 13 1 ----c - --c - - 2400 60 6 Dim of the Quantum subsystem code = 34 2 14 8 --c - --c + -- 75 75 15 Dim of the Quantum stabilizer code = 17 2 7 2 --c - --c + - 30 30 3 Min distance of dual code = 13 2 43 --c + --c - 48 60 5 mindualdistFm(120,11/20, 1/3, 3)-- with c=120=11^2-1 ell= 76 ino = 44 The min height = 41 -- 3 Dim of the Class code = 26879 ----- 6 Dim of the Quantum subsystem code = 97592 ----- 15 Dim of the Quantum stabilizer code = 24398 ----- 3 Min distance of dual code = 4104 IV. mindualdistFm(c, 11/20, 1/3, 4)--- This is a Hirzebruch surface of type F4. ell= 2 -c - 4 3 ino = 2 --c + 4 15 The min height = 1 1 --c - - 30 4 Dim of the Class code = 149 2 3 1 ---c - --c - - 450 10 8 Dim of the Quantum subsystem code = 8 2 2 2 --c + --c + - 25 25 5 Dim of the Quantum stabilizer code = 2 2 1 1 -c + --c + - 5 10 2 Min distance of dual code = 13 2 152 --c + ---c - 64 45 15 mindualdistFm(288, 11/20, 1/3, 4) -- with c=288= 17^2-1 ell= 188 ino = 212 --- 5 The min height = 187 --- 20 Dim of the Class code = 5475431 ------- = 27377 200 Dim of the Quantum subsystem code = 664138 ------ 25 Dim of the Quantum stabilizer code = 332069 ------ 10 Min distance of dual code = 26816