Best Known (40, 40+33, s)-Nets in Base 27
(40, 40+33, 196)-Net over F27 — Constructive and digital
Digital (40, 73, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 15, 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, 20, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (5, 38, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 15, 64)-net over F27, using
(40, 40+33, 370)-Net in Base 27 — Constructive
(40, 73, 370)-net in base 27, using
- 23 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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, 72, 370)-net over F81, using
(40, 40+33, 927)-Net over F27 — Digital
Digital (40, 73, 927)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 927, F27, 33) (dual of [927, 854, 34]-code), using
- 184 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 29 times 0, 1, 49 times 0, 1, 73 times 0) [i] based on linear OA(2762, 732, F27, 33) (dual of [732, 670, 34]-code), using
- construction XX applied to C1 = C([727,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([727,31]) [i] based on
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [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([727,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([727,31]) [i] based on
- 184 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 29 times 0, 1, 49 times 0, 1, 73 times 0) [i] based on linear OA(2762, 732, F27, 33) (dual of [732, 670, 34]-code), using
(40, 40+33, 722266)-Net in Base 27 — Upper bound on s
There is no (40, 73, 722267)-net in base 27, because
- 1 times m-reduction [i] would yield (40, 72, 722267)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 11 433953 213374 211099 229422 237237 766788 924698 209400 128492 630997 642661 195768 865905 969569 073909 268597 875809 > 2772 [i]