Best Known (73−31, 73, s)-Nets in Base 128
(73−31, 73, 1094)-Net over F128 — Constructive and digital
Digital (42, 73, 1094)-net over F128, using
- 1283 times duplication [i] based on digital (39, 70, 1094)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 1094, F128, 31, 31) (dual of [(1094, 31), 33844, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12870, 16411, F128, 31) (dual of [16411, 16341, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16414, F128, 31) (dual of [16414, 16344, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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(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 C([0,15]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16414, F128, 31) (dual of [16414, 16344, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12870, 16411, F128, 31) (dual of [16411, 16341, 32]-code), using
- net defined by OOA [i] based on linear OOA(12870, 1094, F128, 31, 31) (dual of [(1094, 31), 33844, 32]-NRT-code), using
(73−31, 73, 4369)-Net in Base 128 — Constructive
(42, 73, 4369)-net in base 128, using
- 1281 times duplication [i] based on (41, 72, 4369)-net in base 128, using
- base change [i] based on digital (32, 63, 4369)-net over F256, using
- 2562 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on 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]
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- 2562 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- base change [i] based on digital (32, 63, 4369)-net over F256, using
(73−31, 73, 15654)-Net over F128 — Digital
Digital (42, 73, 15654)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12873, 15654, F128, 31) (dual of [15654, 15581, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12873, 16422, F128, 31) (dual of [16422, 16349, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(17) [i] based on
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(30) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12873, 16422, F128, 31) (dual of [16422, 16349, 32]-code), using
(73−31, 73, large)-Net in Base 128 — Upper bound on s
There is no (42, 73, large)-net in base 128, because
- 29 times m-reduction [i] would yield (42, 44, large)-net in base 128, but