Best Known (54−26, 54, s)-Nets in Base 128
(54−26, 54, 1261)-Net over F128 — Constructive and digital
Digital (28, 54, 1261)-net over F128, using
- net defined by OOA [i] based on linear OOA(12854, 1261, F128, 26, 26) (dual of [(1261, 26), 32732, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12854, 16393, F128, 26) (dual of [16393, 16339, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 16395, F128, 26) (dual of [16395, 16341, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 16395, F128, 26) (dual of [16395, 16341, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12854, 16393, F128, 26) (dual of [16393, 16339, 27]-code), using
(54−26, 54, 5459)-Net over F128 — Digital
Digital (28, 54, 5459)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12854, 5459, F128, 3, 26) (dual of [(5459, 3), 16323, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12854, 5465, F128, 3, 26) (dual of [(5465, 3), 16341, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12854, 16395, F128, 26) (dual of [16395, 16341, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(25) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(12854, 16395, F128, 26) (dual of [16395, 16341, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(12854, 5465, F128, 3, 26) (dual of [(5465, 3), 16341, 27]-NRT-code), using
(54−26, 54, large)-Net in Base 128 — Upper bound on s
There is no (28, 54, large)-net in base 128, because
- 24 times m-reduction [i] would yield (28, 30, large)-net in base 128, but