Best Known (53, 53+18, s)-Nets in Base 8
(53, 53+18, 910)-Net over F8 — Constructive and digital
Digital (53, 71, 910)-net over F8, using
- 81 times duplication [i] based on digital (52, 70, 910)-net over F8, using
- net defined by OOA [i] based on linear OOA(870, 910, F8, 18, 18) (dual of [(910, 18), 16310, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(870, 8190, F8, 18) (dual of [8190, 8120, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(870, 8192, F8, 18) (dual of [8192, 8122, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(870, 8190, F8, 18) (dual of [8190, 8120, 19]-code), using
- net defined by OOA [i] based on linear OOA(870, 910, F8, 18, 18) (dual of [(910, 18), 16310, 19]-NRT-code), using
(53, 53+18, 8198)-Net over F8 — Digital
Digital (53, 71, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(871, 8198, F8, 18) (dual of [8198, 8127, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(870, 8196, F8, 18) (dual of [8196, 8126, 19]-code), using
- trace code [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- linear OA(870, 8197, F8, 17) (dual of [8197, 8127, 18]-code), using Gilbert–Varšamov bound and bm = 870 > Vbs−1(k−1) = 649 140869 377289 573886 415146 843877 583999 433758 379323 704041 274880 [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(870, 8196, F8, 18) (dual of [8196, 8126, 19]-code), using
- construction X with Varšamov bound [i] based on
(53, 53+18, 7889140)-Net in Base 8 — Upper bound on s
There is no (53, 71, 7889141)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 13164 046545 910435 569517 854590 425182 276445 552686 790684 879917 092424 > 871 [i]