Best Known (89, 89+15, s)-Nets in Base 8
(89, 89+15, 299622)-Net over F8 — Constructive and digital
Digital (89, 104, 299622)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- digital (77, 92, 299594)-net over F8, using
- net defined by OOA [i] based on linear OOA(892, 299594, F8, 15, 15) (dual of [(299594, 15), 4493818, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(892, 2097159, F8, 15) (dual of [2097159, 2097067, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(885, 2097152, F8, 14) (dual of [2097152, 2097067, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(892, 2097159, F8, 15) (dual of [2097159, 2097067, 16]-code), using
- net defined by OOA [i] based on linear OOA(892, 299594, F8, 15, 15) (dual of [(299594, 15), 4493818, 16]-NRT-code), using
- digital (5, 12, 28)-net over F8, using
(89, 89+15, 599188)-Net in Base 8 — Constructive
(89, 104, 599188)-net in base 8, using
- net defined by OOA [i] based on OOA(8104, 599188, S8, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(8104, 4194317, S8, 15), using
- discarding factors based on OA(8104, 4194320, S8, 15), using
- trace code [i] based on OA(6452, 2097160, S64, 15), using
- discarding parts of the base [i] based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- trace code [i] based on OA(6452, 2097160, S64, 15), using
- discarding factors based on OA(8104, 4194320, S8, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(8104, 4194317, S8, 15), using
(89, 89+15, 4415998)-Net over F8 — Digital
Digital (89, 104, 4415998)-net over F8, using
(89, 89+15, large)-Net in Base 8 — Upper bound on s
There is no (89, 104, large)-net in base 8, because
- 13 times m-reduction [i] would yield (89, 91, large)-net in base 8, but