Best Known (142, 142+26, s)-Nets in Base 8
(142, 142+26, 161328)-Net over F8 — Constructive and digital
Digital (142, 168, 161328)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (129, 155, 161319)-net over F8, using
- net defined by OOA [i] based on linear OOA(8155, 161319, F8, 26, 26) (dual of [(161319, 26), 4194139, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8155, 2097147, F8, 26) (dual of [2097147, 2096992, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8155, 2097147, F8, 26) (dual of [2097147, 2096992, 27]-code), using
- net defined by OOA [i] based on linear OOA(8155, 161319, F8, 26, 26) (dual of [(161319, 26), 4194139, 27]-NRT-code), using
- digital (0, 13, 9)-net over F8, using
(142, 142+26, 2097215)-Net over F8 — Digital
Digital (142, 168, 2097215)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8168, 2097215, F8, 26) (dual of [2097215, 2097047, 27]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8167, 2097213, F8, 26) (dual of [2097213, 2097046, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(812, 61, F8, 7) (dual of [61, 49, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(8167, 2097214, F8, 25) (dual of [2097214, 2097047, 26]-code), using Gilbert–Varšamov bound and bm = 8167 > Vbs−1(k−1) = 16181 241219 326635 259866 871563 018209 161138 236042 045957 291509 572517 292579 079227 935752 357699 177512 889517 203817 430693 758826 598670 697728 289232 771667 743993 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8167, 2097213, F8, 26) (dual of [2097213, 2097046, 27]-code), using
- construction X with Varšamov bound [i] based on
(142, 142+26, large)-Net in Base 8 — Upper bound on s
There is no (142, 168, large)-net in base 8, because
- 24 times m-reduction [i] would yield (142, 144, large)-net in base 8, but