Best Known (53−16, 53, s)-Nets in Base 128
(53−16, 53, 262147)-Net over F128 — Constructive and digital
Digital (37, 53, 262147)-net over F128, using
- 1281 times duplication [i] based on digital (36, 52, 262147)-net over F128, using
- net defined by OOA [i] based on linear OOA(12852, 262147, F128, 16, 16) (dual of [(262147, 16), 4194300, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12852, 2097176, F128, 16) (dual of [2097176, 2097124, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 2097179, F128, 16) (dual of [2097179, 2097127, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(15) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 2097179, F128, 16) (dual of [2097179, 2097127, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12852, 2097176, F128, 16) (dual of [2097176, 2097124, 17]-code), using
- net defined by OOA [i] based on linear OOA(12852, 262147, F128, 16, 16) (dual of [(262147, 16), 4194300, 17]-NRT-code), using
(53−16, 53, 1048575)-Net in Base 128 — Constructive
(37, 53, 1048575)-net in base 128, using
- net defined by OOA [i] based on OOA(12853, 1048575, S128, 16, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
- discarding factors based on OA(12853, large, S128, 16), using
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding parts of the base [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- discarding factors based on OA(12853, large, S128, 16), using
- OA 8-folding and stacking [i] based on OA(12853, 8388600, S128, 16), using
(53−16, 53, 2097183)-Net over F128 — Digital
Digital (37, 53, 2097183)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12853, 2097183, F128, 16) (dual of [2097183, 2097130, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(7) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(15) ⊂ Ce(7) [i] based on
(53−16, 53, large)-Net in Base 128 — Upper bound on s
There is no (37, 53, large)-net in base 128, because
- 14 times m-reduction [i] would yield (37, 39, large)-net in base 128, but