Best Known (97−15, 97, s)-Nets in Base 8
(97−15, 97, 299597)-Net over F8 — Constructive and digital
Digital (82, 97, 299597)-net over F8, using
- 81 times duplication [i] based on digital (81, 96, 299597)-net over F8, using
- net defined by OOA [i] based on linear OOA(896, 299597, F8, 15, 15) (dual of [(299597, 15), 4493859, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(896, 2097180, F8, 15) (dual of [2097180, 2097084, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(896, 2097184, F8, 15) (dual of [2097184, 2097088, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [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(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(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(896, 2097184, F8, 15) (dual of [2097184, 2097088, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(896, 2097180, F8, 15) (dual of [2097180, 2097084, 16]-code), using
- net defined by OOA [i] based on linear OOA(896, 299597, F8, 15, 15) (dual of [(299597, 15), 4493859, 16]-NRT-code), using
(97−15, 97, 2097186)-Net over F8 — Digital
Digital (82, 97, 2097186)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(897, 2097186, F8, 15) (dual of [2097186, 2097089, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(896, 2097184, F8, 15) (dual of [2097184, 2097088, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [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(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(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(896, 2097185, F8, 14) (dual of [2097185, 2097089, 15]-code), using Gilbert–Varšamov bound and bm = 896 > Vbs−1(k−1) = 236185 591452 016500 685730 772280 819524 172285 244952 021044 518434 930124 849045 910454 772973 [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(896, 2097184, F8, 15) (dual of [2097184, 2097088, 16]-code), using
- construction X with Varšamov bound [i] based on
(97−15, 97, large)-Net in Base 8 — Upper bound on s
There is no (82, 97, large)-net in base 8, because
- 13 times m-reduction [i] would yield (82, 84, large)-net in base 8, but