Best Known (26−12, 26, s)-Nets in Base 128
(26−12, 26, 2732)-Net over F128 — Constructive and digital
Digital (14, 26, 2732)-net over F128, using
- 1281 times duplication [i] based on digital (13, 25, 2732)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 2732, F128, 12, 12) (dual of [(2732, 12), 32759, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12825, 16392, F128, 12) (dual of [16392, 16367, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- OA 6-folding and stacking [i] based on linear OA(12825, 16392, F128, 12) (dual of [16392, 16367, 13]-code), using
- net defined by OOA [i] based on linear OOA(12825, 2732, F128, 12, 12) (dual of [(2732, 12), 32759, 13]-NRT-code), using
(26−12, 26, 8197)-Net over F128 — Digital
Digital (14, 26, 8197)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12826, 8197, F128, 2, 12) (dual of [(8197, 2), 16368, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12826, 16394, F128, 12) (dual of [16394, 16368, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 16395, F128, 12) (dual of [16395, 16369, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 16395, F128, 12) (dual of [16395, 16369, 13]-code), using
- OOA 2-folding [i] based on linear OA(12826, 16394, F128, 12) (dual of [16394, 16368, 13]-code), using
(26−12, 26, large)-Net in Base 128 — Upper bound on s
There is no (14, 26, large)-net in base 128, because
- 10 times m-reduction [i] would yield (14, 16, large)-net in base 128, but