Best Known (43, 43+36, s)-Nets in Base 27
(43, 43+36, 196)-Net over F27 — Constructive and digital
Digital (43, 79, 196)-net over F27, using
- 1 times m-reduction [i] based on digital (43, 80, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 16, 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, 22, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (5, 42, 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, 16, 64)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(43, 43+36, 370)-Net in Base 27 — Constructive
(43, 79, 370)-net in base 27, using
- 29 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
(43, 43+36, 930)-Net over F27 — Digital
Digital (43, 79, 930)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2779, 930, F27, 36) (dual of [930, 851, 37]-code), using
- 187 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 30 times 0, 1, 51 times 0, 1, 70 times 0) [i] based on linear OA(2768, 732, F27, 36) (dual of [732, 664, 37]-code), using
- construction XX applied to C1 = C([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [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([727,33]), C2 = C([0,34]), C3 = C1 + C2 = C([0,33]), and C∩ = C1 ∩ C2 = C([727,34]) [i] based on
- 187 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 17 times 0, 1, 30 times 0, 1, 51 times 0, 1, 70 times 0) [i] based on linear OA(2768, 732, F27, 36) (dual of [732, 664, 37]-code), using
(43, 43+36, 556219)-Net in Base 27 — Upper bound on s
There is no (43, 79, 556220)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 119604 424746 140150 191418 661410 283691 512127 637715 913837 719374 718058 475812 975712 965850 690624 558228 022056 088903 835273 > 2779 [i]