Best Known (33, 33+26, s)-Nets in Base 27
(33, 33+26, 192)-Net over F27 — Constructive and digital
Digital (33, 59, 192)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 12, 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, 17, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (4, 30, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (4, 12, 64)-net over F27, using
(33, 33+26, 370)-Net in Base 27 — Constructive
(33, 59, 370)-net in base 27, using
- 9 times m-reduction [i] based on (33, 68, 370)-net in base 27, using
- base change [i] based on digital (16, 51, 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, 51, 370)-net over F81, using
(33, 33+26, 949)-Net over F27 — Digital
Digital (33, 59, 949)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2759, 949, F27, 26) (dual of [949, 890, 27]-code), using
- 209 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 9 times 0, 1, 28 times 0, 1, 63 times 0, 1, 102 times 0) [i] based on linear OA(2751, 732, F27, 26) (dual of [732, 681, 27]-code), using
- construction XX applied to C1 = C([727,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([727,24]) [i] based on
- linear OA(2749, 728, F27, 25) (dual of [728, 679, 26]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2749, 728, F27, 25) (dual of [728, 679, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2751, 728, F27, 26) (dual of [728, 677, 27]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,24}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [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([727,23]), C2 = C([0,24]), C3 = C1 + C2 = C([0,23]), and C∩ = C1 ∩ C2 = C([727,24]) [i] based on
- 209 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 9 times 0, 1, 28 times 0, 1, 63 times 0, 1, 102 times 0) [i] based on linear OA(2751, 732, F27, 26) (dual of [732, 681, 27]-code), using
(33, 33+26, 683310)-Net in Base 27 — Upper bound on s
There is no (33, 59, 683311)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 2 821410 446304 694234 358948 794402 849918 358314 636465 935963 061233 378393 613979 394670 907919 > 2759 [i]