Best Known (40, 40+37, s)-Nets in Base 128
(40, 40+37, 910)-Net over F128 — Constructive and digital
Digital (40, 77, 910)-net over F128, using
- 1284 times duplication [i] based on digital (36, 73, 910)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(12873, 16381, F128, 37) (dual of [16381, 16308, 38]-code), using
- net defined by OOA [i] based on linear OOA(12873, 910, F128, 37, 37) (dual of [(910, 37), 33597, 38]-NRT-code), using
(40, 40+37, 5466)-Net over F128 — Digital
Digital (40, 77, 5466)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12877, 5466, F128, 3, 37) (dual of [(5466, 3), 16321, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12877, 16398, F128, 37) (dual of [16398, 16321, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- OOA 3-folding [i] based on linear OA(12877, 16398, F128, 37) (dual of [16398, 16321, 38]-code), using
(40, 40+37, large)-Net in Base 128 — Upper bound on s
There is no (40, 77, large)-net in base 128, because
- 35 times m-reduction [i] would yield (40, 42, large)-net in base 128, but