Best Known (52, 52+45, s)-Nets in Base 27
(52, 52+45, 210)-Net over F27 — Constructive and digital
Digital (52, 97, 210)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 19, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (4, 26, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (7, 52, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (4, 19, 64)-net over F27, using
(52, 52+45, 370)-Net in Base 27 — Constructive
(52, 97, 370)-net in base 27, using
- t-expansion [i] based on (43, 97, 370)-net in base 27, using
- 11 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- 11 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(52, 52+45, 972)-Net over F27 — Digital
Digital (52, 97, 972)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2797, 972, F27, 45) (dual of [972, 875, 46]-code), using
- 229 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 6 times 0, 1, 14 times 0, 1, 28 times 0, 1, 46 times 0, 1, 59 times 0, 1, 67 times 0) [i] based on linear OA(2786, 732, F27, 45) (dual of [732, 646, 46]-code), using
- construction XX applied to C1 = C([727,42]), C2 = C([0,43]), C3 = C1 + C2 = C([0,42]), and C∩ = C1 ∩ C2 = C([727,43]) [i] based on
- linear OA(2784, 728, F27, 44) (dual of [728, 644, 45]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,42}, and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2784, 728, F27, 44) (dual of [728, 644, 45]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,43], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2786, 728, F27, 45) (dual of [728, 642, 46]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,43}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(2782, 728, F27, 43) (dual of [728, 646, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,42]), C2 = C([0,43]), C3 = C1 + C2 = C([0,42]), and C∩ = C1 ∩ C2 = C([727,43]) [i] based on
- 229 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 6 times 0, 1, 14 times 0, 1, 28 times 0, 1, 46 times 0, 1, 59 times 0, 1, 67 times 0) [i] based on linear OA(2786, 732, F27, 45) (dual of [732, 646, 46]-code), using
(52, 52+45, 613511)-Net in Base 27 — Upper bound on s
There is no (52, 97, 613512)-net in base 27, because
- 1 times m-reduction [i] would yield (52, 96, 613512)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 257591 256273 424532 261674 132928 102185 913944 732083 491429 191535 854244 201746 034090 548900 685769 410521 645093 433002 084091 999917 910350 430541 598257 > 2796 [i]