Best Known (40, 72, s)-Nets in Base 128
(40, 72, 1025)-Net over F128 — Constructive and digital
Digital (40, 72, 1025)-net over F128, using
- 1282 times duplication [i] based on digital (38, 70, 1025)-net over F128, using
- t-expansion [i] based on digital (37, 70, 1025)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 1025, F128, 33, 33) (dual of [(1025, 33), 33755, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12870, 16401, F128, 33) (dual of [16401, 16331, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [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(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16402, F128, 33) (dual of [16402, 16332, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(12870, 16401, F128, 33) (dual of [16401, 16331, 34]-code), using
- net defined by OOA [i] based on linear OOA(12870, 1025, F128, 33, 33) (dual of [(1025, 33), 33755, 34]-NRT-code), using
- t-expansion [i] based on digital (37, 70, 1025)-net over F128, using
(40, 72, 4096)-Net in Base 128 — Constructive
(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
(40, 72, 9191)-Net over F128 — Digital
Digital (40, 72, 9191)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12872, 9191, F128, 32) (dual of [9191, 9119, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 16413, F128, 32) (dual of [16413, 16341, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(21) [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(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(1289, 29, F128, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to Ce(31) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 16413, F128, 32) (dual of [16413, 16341, 33]-code), using
(40, 72, 10923)-Net in Base 128
(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
(40, 72, large)-Net in Base 128 — Upper bound on s
There is no (40, 72, large)-net in base 128, because
- 30 times m-reduction [i] would yield (40, 42, large)-net in base 128, but