Best Known (56−17, 56, s)-Nets in Base 128
(56−17, 56, 262147)-Net over F128 — Constructive and digital
Digital (39, 56, 262147)-net over F128, using
- 1281 times duplication [i] based on digital (38, 55, 262147)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 262147, F128, 17, 17) (dual of [(262147, 17), 4456444, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12855, 2097177, F128, 17) (dual of [2097177, 2097122, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 2097179, F128, 17) (dual of [2097179, 2097124, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1286, 27, F128, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12855, 2097179, F128, 17) (dual of [2097179, 2097124, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12855, 2097177, F128, 17) (dual of [2097177, 2097122, 18]-code), using
- net defined by OOA [i] based on linear OOA(12855, 262147, F128, 17, 17) (dual of [(262147, 17), 4456444, 18]-NRT-code), using
(56−17, 56, 1048575)-Net in Base 128 — Constructive
(39, 56, 1048575)-net in base 128, using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
(56−17, 56, 2097184)-Net over F128 — Digital
Digital (39, 56, 2097184)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12856, 2097184, F128, 17) (dual of [2097184, 2097128, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,4]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,8]) ⊂ C([0,4]) [i] based on
(56−17, 56, 2882188)-Net in Base 128
(39, 56, 2882188)-net in base 128, using
- base change [i] based on digital (32, 49, 2882188)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 2882188, F256, 2, 17) (dual of [(2882188, 2), 5764327, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25649, 2882188, F256, 2, 17) (dual of [(2882188, 2), 5764327, 18]-NRT-code), using
(56−17, 56, large)-Net in Base 128 — Upper bound on s
There is no (39, 56, large)-net in base 128, because
- 15 times m-reduction [i] would yield (39, 41, large)-net in base 128, but