Best Known (103−29, 103, s)-Nets in Base 8
(103−29, 103, 514)-Net over F8 — Constructive and digital
Digital (74, 103, 514)-net over F8, using
- 81 times duplication [i] based on digital (73, 102, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 30, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 15, 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, 15, 80)-net over F64, using
- digital (43, 72, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 177)-net over F64, using
- digital (16, 30, 160)-net over F8, using
- (u, u+v)-construction [i] based on
(103−29, 103, 576)-Net in Base 8 — Constructive
(74, 103, 576)-net in base 8, using
- t-expansion [i] based on (73, 103, 576)-net in base 8, using
- 9 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
- 9 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
(103−29, 103, 4011)-Net over F8 — Digital
Digital (74, 103, 4011)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8103, 4011, F8, 29) (dual of [4011, 3908, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8103, 4107, F8, 29) (dual of [4107, 4004, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(8101, 4096, F8, 29) (dual of [4096, 3995, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(893, 4096, F8, 27) (dual of [4096, 4003, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(889, 4096, F8, 26) (dual of [4096, 4007, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(81, 10, F8, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8103, 4107, F8, 29) (dual of [4107, 4004, 30]-code), using
(103−29, 103, 3281060)-Net in Base 8 — Upper bound on s
There is no (74, 103, 3281061)-net in base 8, because
- 1 times m-reduction [i] would yield (74, 102, 3281061)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 130 370768 254365 395306 814061 816360 707482 140446 694604 606874 555980 476467 224209 764189 925116 090032 > 8102 [i]