Best Known (133, 133+24, s)-Nets in Base 8
(133, 133+24, 174766)-Net over F8 — Constructive and digital
Digital (133, 157, 174766)-net over F8, using
- 82 times duplication [i] based on digital (131, 155, 174766)-net over F8, using
- t-expansion [i] based on digital (130, 155, 174766)-net over F8, using
- net defined by OOA [i] based on linear OOA(8155, 174766, F8, 25, 25) (dual of [(174766, 25), 4368995, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8155, 2097193, F8, 25) (dual of [2097193, 2097038, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8155, 2097194, F8, 25) (dual of [2097194, 2097039, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8113, 2097152, F8, 19) (dual of [2097152, 2097039, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(24) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8155, 2097194, F8, 25) (dual of [2097194, 2097039, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8155, 2097193, F8, 25) (dual of [2097193, 2097038, 26]-code), using
- net defined by OOA [i] based on linear OOA(8155, 174766, F8, 25, 25) (dual of [(174766, 25), 4368995, 26]-NRT-code), using
- t-expansion [i] based on digital (130, 155, 174766)-net over F8, using
(133, 133+24, 2097205)-Net over F8 — Digital
Digital (133, 157, 2097205)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8157, 2097205, F8, 24) (dual of [2097205, 2097048, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8155, 2097201, F8, 24) (dual of [2097201, 2097046, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 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(87, 49, F8, 5) (dual of [49, 42, 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(24) ⊂ Ce(17) [i] based on
- linear OA(8155, 2097203, F8, 23) (dual of [2097203, 2097048, 24]-code), using Gilbert–Varšamov bound and bm = 8155 > Vbs−1(k−1) = 41440 872736 745127 272002 774503 496160 501227 956327 177782 565907 077514 283975 281604 543243 517213 130336 527517 439568 024077 374150 980318 972194 693584 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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(8155, 2097201, F8, 24) (dual of [2097201, 2097046, 25]-code), using
- construction X with Varšamov bound [i] based on
(133, 133+24, large)-Net in Base 8 — Upper bound on s
There is no (133, 157, large)-net in base 8, because
- 22 times m-reduction [i] would yield (133, 135, large)-net in base 8, but