Best Known (73−32, 73, s)-Nets in Base 128
(73−32, 73, 1026)-Net over F128 — Constructive and digital
Digital (41, 73, 1026)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 1026, F128, 32, 32) (dual of [(1026, 32), 32759, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(20) [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(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(31) ⊂ Ce(20) [i] based on
- OA 16-folding and stacking [i] based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
(73−32, 73, 4096)-Net in Base 128 — Constructive
(41, 73, 4096)-net in base 128, using
- 1281 times duplication [i] based on (40, 72, 4096)-net in base 128, using
- base change [i] based on digital (31, 63, 4096)-net over F256, using
- net defined by OOA [i] based on linear OOA(25663, 4096, F256, 32, 32) (dual of [(4096, 32), 131009, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on 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]
- OA 16-folding and stacking [i] based on linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using
- net defined by OOA [i] based on linear OOA(25663, 4096, F256, 32, 32) (dual of [(4096, 32), 131009, 33]-NRT-code), using
- base change [i] based on digital (31, 63, 4096)-net over F256, using
(73−32, 73, 10807)-Net over F128 — Digital
Digital (41, 73, 10807)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12873, 10807, F128, 32) (dual of [10807, 10734, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(20) [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(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(31) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
(73−32, 73, 10923)-Net in Base 128
(41, 73, 10923)-net in base 128, using
- 1281 times duplication [i] based on (40, 72, 10923)-net in base 128, using
- base change [i] based on digital (31, 63, 10923)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25663, 10923, F256, 6, 32) (dual of [(10923, 6), 65475, 33]-NRT-code), using
- OOA 6-folding [i] based on linear OA(25663, 65538, F256, 32) (dual of [65538, 65475, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- 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(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31) ⊂ Ce(30) [i] based on
- OOA 6-folding [i] based on linear OA(25663, 65538, F256, 32) (dual of [65538, 65475, 33]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25663, 10923, F256, 6, 32) (dual of [(10923, 6), 65475, 33]-NRT-code), using
- base change [i] based on digital (31, 63, 10923)-net over F256, using
(73−32, 73, large)-Net in Base 128 — Upper bound on s
There is no (41, 73, large)-net in base 128, because
- 30 times m-reduction [i] would yield (41, 43, large)-net in base 128, but