Best Known (29−13, 29, s)-Nets in Base 128
(29−13, 29, 2732)-Net over F128 — Constructive and digital
Digital (16, 29, 2732)-net over F128, using
- 1281 times duplication [i] based on digital (15, 28, 2732)-net over F128, using
- net defined by OOA [i] based on linear OOA(12828, 2732, F128, 13, 13) (dual of [(2732, 13), 35488, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12828, 16393, F128, 13) (dual of [16393, 16365, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12828, 16396, F128, 13) (dual of [16396, 16368, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12828, 16396, F128, 13) (dual of [16396, 16368, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12828, 16393, F128, 13) (dual of [16393, 16365, 14]-code), using
- net defined by OOA [i] based on linear OOA(12828, 2732, F128, 13, 13) (dual of [(2732, 13), 35488, 14]-NRT-code), using
(29−13, 29, 8929)-Net over F128 — Digital
Digital (16, 29, 8929)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12829, 8929, F128, 13) (dual of [8929, 8900, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12829, 16398, F128, 13) (dual of [16398, 16369, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12829, 16398, F128, 13) (dual of [16398, 16369, 14]-code), using
(29−13, 29, 10922)-Net in Base 128 — Constructive
(16, 29, 10922)-net in base 128, using
- net defined by OOA [i] based on OOA(12829, 10922, S128, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12829, 65533, S128, 13), using
- discarding factors based on OA(12829, 65538, S128, 13), using
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- discarding factors based on OA(12829, 65538, S128, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12829, 65533, S128, 13), using
(29−13, 29, large)-Net in Base 128 — Upper bound on s
There is no (16, 29, large)-net in base 128, because
- 11 times m-reduction [i] would yield (16, 18, large)-net in base 128, but