Best Known (44−20, 44, s)-Nets in Base 27
(44−20, 44, 164)-Net over F27 — Constructive and digital
Digital (24, 44, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 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 (7, 27, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 17, 82)-net over F27, using
(44−20, 44, 224)-Net in Base 27 — Constructive
(24, 44, 224)-net in base 27, using
- base change [i] based on digital (13, 33, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
(44−20, 44, 755)-Net over F27 — Digital
Digital (24, 44, 755)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2744, 755, F27, 20) (dual of [755, 711, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2744, 762, F27, 20) (dual of [762, 718, 21]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 17 times 0) [i] based on linear OA(2739, 732, F27, 20) (dual of [732, 693, 21]-code), using
- construction XX applied to C1 = C([727,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([727,18]) [i] based on
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [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([727,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([727,18]) [i] based on
- 25 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 4 times 0, 1, 17 times 0) [i] based on linear OA(2739, 732, F27, 20) (dual of [732, 693, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2744, 762, F27, 20) (dual of [762, 718, 21]-code), using
(44−20, 44, 345937)-Net in Base 27 — Upper bound on s
There is no (24, 44, 345938)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 955 024407 246032 368960 637777 434347 186821 585833 775695 688746 082053 > 2744 [i]