Best Known (114−18, 114, s)-Nets in Base 8
(114−18, 114, 233021)-Net over F8 — Constructive and digital
Digital (96, 114, 233021)-net over F8, using
- 81 times duplication [i] based on digital (95, 113, 233021)-net over F8, using
- net defined by OOA [i] based on linear OOA(8113, 233021, F8, 18, 18) (dual of [(233021, 18), 4194265, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8113, 2097189, F8, 18) (dual of [2097189, 2097076, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 2097194, F8, 18) (dual of [2097194, 2097081, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [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(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(8113, 2097194, F8, 18) (dual of [2097194, 2097081, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8113, 2097189, F8, 18) (dual of [2097189, 2097076, 19]-code), using
- net defined by OOA [i] based on linear OOA(8113, 233021, F8, 18, 18) (dual of [(233021, 18), 4194265, 19]-NRT-code), using
(114−18, 114, 2097196)-Net over F8 — Digital
Digital (96, 114, 2097196)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8114, 2097196, F8, 18) (dual of [2097196, 2097082, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8113, 2097194, F8, 18) (dual of [2097194, 2097081, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [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(871, 2097152, F8, 12) (dual of [2097152, 2097081, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(8113, 2097195, F8, 17) (dual of [2097195, 2097082, 18]-code), using Gilbert–Varšamov bound and bm = 8113 > Vbs−1(k−1) = 222403 887578 561040 312467 217876 825338 896621 996501 283819 858536 510723 730367 809214 341161 413392 825129 410866 [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(8113, 2097194, F8, 18) (dual of [2097194, 2097081, 19]-code), using
- construction X with Varšamov bound [i] based on
(114−18, 114, large)-Net in Base 8 — Upper bound on s
There is no (96, 114, large)-net in base 8, because
- 16 times m-reduction [i] would yield (96, 98, large)-net in base 8, but