Best Known (99, 99+18, s)-Nets in Base 8
(99, 99+18, 233034)-Net over F8 — Constructive and digital
Digital (99, 117, 233034)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- digital (88, 106, 233017)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
- digital (2, 11, 17)-net over F8, using
(99, 99+18, 2097206)-Net over F8 — Digital
Digital (99, 117, 2097206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 2097206, F8, 18) (dual of [2097206, 2097089, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8116, 2097204, F8, 18) (dual of [2097204, 2097088, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- 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(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(810, 52, F8, 6) (dual of [52, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(8116, 2097205, F8, 17) (dual of [2097205, 2097089, 18]-code), using Gilbert–Varšamov bound and bm = 8116 > Vbs−1(k−1) = 222420 855975 125301 125171 722968 421660 507470 942101 409441 101012 939545 927363 596226 634562 660654 648257 896231 [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(8116, 2097204, F8, 18) (dual of [2097204, 2097088, 19]-code), using
- construction X with Varšamov bound [i] based on
(99, 99+18, large)-Net in Base 8 — Upper bound on s
There is no (99, 117, large)-net in base 8, because
- 16 times m-reduction [i] would yield (99, 101, large)-net in base 8, but