Best Known (44, 44+33, s)-Nets in Base 128
(44, 44+33, 1026)-Net over F128 — Constructive and digital
Digital (44, 77, 1026)-net over F128, using
- 1281 times duplication [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
(44, 44+33, 4096)-Net in Base 128 — Constructive
(44, 77, 4096)-net in base 128, using
- 1282 times duplication [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
(44, 44+33, 14317)-Net over F128 — Digital
Digital (44, 77, 14317)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12877, 14317, F128, 33) (dual of [14317, 14240, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 16422, F128, 33) (dual of [16422, 16345, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(19) [i] based on
- linear OA(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(32) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12877, 16422, F128, 33) (dual of [16422, 16345, 34]-code), using
(44, 44+33, large)-Net in Base 128 — Upper bound on s
There is no (44, 77, large)-net in base 128, because
- 31 times m-reduction [i] would yield (44, 46, large)-net in base 128, but