Best Known (41−13, 41, s)-Nets in Base 128
(41−13, 41, 349528)-Net over F128 — Constructive and digital
Digital (28, 41, 349528)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 349528, F128, 13, 13) (dual of [(349528, 13), 4543823, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12841, 2097169, F128, 13) (dual of [2097169, 2097128, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12841, 2097169, F128, 13) (dual of [2097169, 2097128, 14]-code), using
(41−13, 41, 1777424)-Net over F128 — Digital
Digital (28, 41, 1777424)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12841, 1777424, F128, 13) (dual of [1777424, 1777383, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
(41−13, 41, large)-Net in Base 128 — Upper bound on s
There is no (28, 41, large)-net in base 128, because
- 11 times m-reduction [i] would yield (28, 30, large)-net in base 128, but