Best Known (167−24, 167, s)-Nets in Base 8
(167−24, 167, 174797)-Net over F8 — Constructive and digital
Digital (143, 167, 174797)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (8, 20, 35)-net over F8, using
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 7, N(F) = 34, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 7 and N(F) ≥ 34, using a function field by Sémirat [i]
- net from sequence [i] based on digital (8, 34)-sequence over F8, using
- digital (123, 147, 174762)-net over F8, using
- net defined by OOA [i] based on linear OOA(8147, 174762, F8, 24, 24) (dual of [(174762, 24), 4194141, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8147, 2097144, F8, 24) (dual of [2097144, 2096997, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 2097151, F8, 24) (dual of [2097151, 2097004, 25]-code), using
- 1 times truncation [i] based on linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 1 times truncation [i] based on linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 2097151, F8, 24) (dual of [2097151, 2097004, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(8147, 2097144, F8, 24) (dual of [2097144, 2096997, 25]-code), using
- net defined by OOA [i] based on linear OOA(8147, 174762, F8, 24, 24) (dual of [(174762, 24), 4194141, 25]-NRT-code), using
- digital (8, 20, 35)-net over F8, using
(167−24, 167, 349526)-Net in Base 8 — Constructive
(143, 167, 349526)-net in base 8, using
- 81 times duplication [i] based on (142, 166, 349526)-net in base 8, using
- net defined by OOA [i] based on OOA(8166, 349526, S8, 24, 24), using
- OA 12-folding and stacking [i] based on OA(8166, 4194312, S8, 24), using
- discarding factors based on OA(8166, 4194318, S8, 24), using
- trace code [i] based on OA(6483, 2097159, S64, 24), using
- discarding parts of the base [i] based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(12871, 2097159, F128, 24) (dual of [2097159, 2097088, 25]-code), using
- trace code [i] based on OA(6483, 2097159, S64, 24), using
- discarding factors based on OA(8166, 4194318, S8, 24), using
- OA 12-folding and stacking [i] based on OA(8166, 4194312, S8, 24), using
- net defined by OOA [i] based on OOA(8166, 349526, S8, 24, 24), using
(167−24, 167, 4859346)-Net over F8 — Digital
Digital (143, 167, 4859346)-net over F8, using
(167−24, 167, large)-Net in Base 8 — Upper bound on s
There is no (143, 167, large)-net in base 8, because
- 22 times m-reduction [i] would yield (143, 145, large)-net in base 8, but