Best Known (83, 83+33, s)-Nets in Base 8
(83, 83+33, 514)-Net over F8 — Constructive and digital
Digital (83, 116, 514)-net over F8, using
- 82 times duplication [i] based on digital (81, 114, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (18, 34, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 17, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 17, 80)-net over F64, using
- digital (47, 80, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- digital (18, 34, 160)-net over F8, using
- (u, u+v)-construction [i] based on
(83, 83+33, 593)-Net in Base 8 — Constructive
(83, 116, 593)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (2, 18, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- (65, 98, 576)-net in base 8, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 42, 288)-net over F128, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
- digital (2, 18, 17)-net over F8, using
(83, 83+33, 3954)-Net over F8 — Digital
Digital (83, 116, 3954)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8116, 3954, F8, 33) (dual of [3954, 3838, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 4107, F8, 33) (dual of [4107, 3991, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(8105, 4097, F8, 29) (dual of [4097, 3992, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(83, 10, F8, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8116, 4107, F8, 33) (dual of [4107, 3991, 34]-code), using
(83, 83+33, 3008836)-Net in Base 8 — Upper bound on s
There is no (83, 116, 3008837)-net in base 8, because
- 1 times m-reduction [i] would yield (83, 115, 3008837)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 71 672115 274088 034177 386792 578184 306466 902470 428855 339637 219291 227811 811910 931254 525258 643860 830035 330140 > 8115 [i]