Best Known (63, 88, s)-Nets in Base 8
(63, 88, 484)-Net over F8 — Constructive and digital
Digital (63, 88, 484)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
- digital (39, 64, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 32, 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, 32, 177)-net over F64, using
- digital (12, 24, 130)-net over F8, using
(63, 88, 576)-Net in Base 8 — Constructive
(63, 88, 576)-net in base 8, using
- 6 times m-reduction [i] based on (63, 94, 576)-net in base 8, using
- trace code for nets [i] based on (16, 47, 288)-net in base 64, using
- 2 times m-reduction [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
- 2 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- trace code for nets [i] based on (16, 47, 288)-net in base 64, using
(63, 88, 3497)-Net over F8 — Digital
Digital (63, 88, 3497)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(888, 3497, F8, 25) (dual of [3497, 3409, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(888, 4107, F8, 25) (dual of [4107, 4019, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(885, 4096, F8, 25) (dual of [4096, 4011, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(877, 4096, F8, 22) (dual of [4096, 4019, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(83, 11, F8, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(888, 4107, F8, 25) (dual of [4107, 4019, 26]-code), using
(63, 88, 2664799)-Net in Base 8 — Upper bound on s
There is no (63, 88, 2664800)-net in base 8, because
- 1 times m-reduction [i] would yield (63, 87, 2664800)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 3 705348 501323 589732 884603 534915 429516 981438 017818 224560 017021 077751 536541 891461 > 887 [i]