Best Known (39−13, 39, s)-Nets in Base 128
(39−13, 39, 349527)-Net over F128 — Constructive and digital
Digital (26, 39, 349527)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 349527, F128, 13, 13) (dual of [(349527, 13), 4543812, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [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(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(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-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(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
(39−13, 39, 1048581)-Net over F128 — Digital
Digital (26, 39, 1048581)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12839, 1048581, F128, 2, 13) (dual of [(1048581, 2), 2097123, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12839, 2097162, F128, 13) (dual of [2097162, 2097123, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [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(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(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- OOA 2-folding [i] based on linear OA(12839, 2097162, F128, 13) (dual of [2097162, 2097123, 14]-code), using
(39−13, 39, large)-Net in Base 128 — Upper bound on s
There is no (26, 39, large)-net in base 128, because
- 11 times m-reduction [i] would yield (26, 28, large)-net in base 128, but