Best Known (75−32, 75, s)-Nets in Base 128
(75−32, 75, 1026)-Net over F128 — Constructive and digital
Digital (43, 75, 1026)-net over F128, using
- 1 times m-reduction [i] based on digital (43, 76, 1026)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 1026, F128, 33, 33) (dual of [(1026, 33), 33782, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12876, 16417, F128, 33) (dual of [16417, 16341, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 16420, F128, 33) (dual of [16420, 16344, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,10]) [i] based on
- linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12811, 35, F128, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,128)), using
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- Reed–Solomon code RS(117,128) [i]
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- construction X applied to C([0,16]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12876, 16420, F128, 33) (dual of [16420, 16344, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12876, 16417, F128, 33) (dual of [16417, 16341, 34]-code), using
- net defined by OOA [i] based on linear OOA(12876, 1026, F128, 33, 33) (dual of [(1026, 33), 33782, 34]-NRT-code), using
(75−32, 75, 4096)-Net in Base 128 — Constructive
(43, 75, 4096)-net in base 128, using
- t-expansion [i] based on (42, 75, 4096)-net in base 128, using
- net defined by OOA [i] based on OOA(12875, 4096, S128, 33, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(12875, 65537, S128, 33), using
- discarding factors based on OA(12875, 65538, S128, 33), using
- discarding parts of the base [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- linear OA(25665, 65536, F256, 33) (dual of [65536, 65471, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(32) ⊂ Ce(31) [i] based on
- discarding parts of the base [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
- discarding factors based on OA(12875, 65538, S128, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(12875, 65537, S128, 33), using
- net defined by OOA [i] based on OOA(12875, 4096, S128, 33, 33), using
(75−32, 75, 14940)-Net over F128 — Digital
Digital (43, 75, 14940)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12875, 14940, F128, 32) (dual of [14940, 14865, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12875, 16422, F128, 32) (dual of [16422, 16347, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(18) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12812, 38, F128, 12) (dual of [38, 26, 13]-code or 38-arc in PG(11,128)), using
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- Reed–Solomon code RS(116,128) [i]
- discarding factors / shortening the dual code based on linear OA(12812, 128, F128, 12) (dual of [128, 116, 13]-code or 128-arc in PG(11,128)), using
- construction X applied to Ce(31) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12875, 16422, F128, 32) (dual of [16422, 16347, 33]-code), using
(75−32, 75, large)-Net in Base 128 — Upper bound on s
There is no (43, 75, large)-net in base 128, because
- 30 times m-reduction [i] would yield (43, 45, large)-net in base 128, but