Best Known (54−24, 54, s)-Nets in Base 128
(54−24, 54, 1367)-Net over F128 — Constructive and digital
Digital (30, 54, 1367)-net over F128, using
- 1281 times duplication [i] based on digital (29, 53, 1367)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1367, F128, 24, 24) (dual of [(1367, 24), 32755, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12853, 16404, F128, 24) (dual of [16404, 16351, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(16) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(23) ⊂ Ce(16) [i] based on
- OA 12-folding and stacking [i] based on linear OA(12853, 16404, F128, 24) (dual of [16404, 16351, 25]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1367, F128, 24, 24) (dual of [(1367, 24), 32755, 25]-NRT-code), using
(54−24, 54, 5461)-Net in Base 128 — Constructive
(30, 54, 5461)-net in base 128, using
- net defined by OOA [i] based on OOA(12854, 5461, S128, 24, 24), using
- OA 12-folding and stacking [i] based on OA(12854, 65532, S128, 24), using
- discarding factors based on OA(12854, 65538, S128, 24), using
- discarding parts of the base [i] based on linear OA(25647, 65538, F256, 24) (dual of [65538, 65491, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- discarding parts of the base [i] based on linear OA(25647, 65538, F256, 24) (dual of [65538, 65491, 25]-code), using
- discarding factors based on OA(12854, 65538, S128, 24), using
- OA 12-folding and stacking [i] based on OA(12854, 65532, S128, 24), using
(54−24, 54, 8492)-Net over F128 — Digital
Digital (30, 54, 8492)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12854, 8492, F128, 24) (dual of [8492, 8438, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 16407, F128, 24) (dual of [16407, 16353, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 16407, F128, 24) (dual of [16407, 16353, 25]-code), using
(54−24, 54, large)-Net in Base 128 — Upper bound on s
There is no (30, 54, large)-net in base 128, because
- 22 times m-reduction [i] would yield (30, 32, large)-net in base 128, but