Best Known (50−22, 50, s)-Nets in Base 128
(50−22, 50, 1491)-Net over F128 — Constructive and digital
Digital (28, 50, 1491)-net over F128, using
- t-expansion [i] based on digital (27, 50, 1491)-net over F128, using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
(50−22, 50, 5958)-Net in Base 128 — Constructive
(28, 50, 5958)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 5958, S128, 22, 22), using
- OA 11-folding and stacking [i] based on OA(12850, 65538, S128, 22), using
- discarding parts of the base [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- OA 11-folding and stacking [i] based on OA(12850, 65538, S128, 22), using
(50−22, 50, 9501)-Net over F128 — Digital
Digital (28, 50, 9501)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12850, 9501, F128, 22) (dual of [9501, 9451, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12850, 16407, F128, 22) (dual of [16407, 16357, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(13) [i] based on
- 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(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(21) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12850, 16407, F128, 22) (dual of [16407, 16357, 23]-code), using
(50−22, 50, large)-Net in Base 128 — Upper bound on s
There is no (28, 50, large)-net in base 128, because
- 20 times m-reduction [i] would yield (28, 30, large)-net in base 128, but