Best Known (44−14, 44, s)-Nets in Base 128
(44−14, 44, 299595)-Net over F128 — Constructive and digital
Digital (30, 44, 299595)-net over F128, using
- 1281 times duplication [i] based on digital (29, 43, 299595)-net over F128, using
- net defined by OOA [i] based on linear OOA(12843, 299595, F128, 14, 14) (dual of [(299595, 14), 4194287, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12843, 2097165, F128, 14) (dual of [2097165, 2097122, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12843, 2097167, F128, 14) (dual of [2097167, 2097124, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12843, 2097167, F128, 14) (dual of [2097167, 2097124, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12843, 2097165, F128, 14) (dual of [2097165, 2097122, 15]-code), using
- net defined by OOA [i] based on linear OOA(12843, 299595, F128, 14, 14) (dual of [(299595, 14), 4194287, 15]-NRT-code), using
(44−14, 44, 1480444)-Net over F128 — Digital
Digital (30, 44, 1480444)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12844, 1480444, F128, 14) (dual of [1480444, 1480400, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12844, 2097171, F128, 14) (dual of [2097171, 2097127, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-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(13) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12844, 2097171, F128, 14) (dual of [2097171, 2097127, 15]-code), using
(44−14, 44, large)-Net in Base 128 — Upper bound on s
There is no (30, 44, large)-net in base 128, because
- 12 times m-reduction [i] would yield (30, 32, large)-net in base 128, but