Best Known (21, 37, s)-Nets in Base 128
(21, 37, 2050)-Net over F128 — Constructive and digital
Digital (21, 37, 2050)-net over F128, using
- 1 times m-reduction [i] based on digital (21, 38, 2050)-net over F128, using
- net defined by OOA [i] based on linear OOA(12838, 2050, F128, 17, 17) (dual of [(2050, 17), 34812, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12838, 16401, F128, 17) (dual of [16401, 16363, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12838, 16402, F128, 17) (dual of [16402, 16364, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12838, 16402, F128, 17) (dual of [16402, 16364, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12838, 16401, F128, 17) (dual of [16401, 16363, 18]-code), using
- net defined by OOA [i] based on linear OOA(12838, 2050, F128, 17, 17) (dual of [(2050, 17), 34812, 18]-NRT-code), using
(21, 37, 8192)-Net in Base 128 — Constructive
(21, 37, 8192)-net in base 128, using
- 1 times m-reduction [i] based on (21, 38, 8192)-net in base 128, using
- net defined by OOA [i] based on OOA(12838, 8192, S128, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(12838, 65537, S128, 17), using
- discarding factors based on OA(12838, 65538, S128, 17), using
- discarding parts of the base [i] based on linear OA(25633, 65538, F256, 17) (dual of [65538, 65505, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(25633, 65538, F256, 17) (dual of [65538, 65505, 18]-code), using
- discarding factors based on OA(12838, 65538, S128, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(12838, 65537, S128, 17), using
- net defined by OOA [i] based on OOA(12838, 8192, S128, 17, 17), using
(21, 37, 12473)-Net over F128 — Digital
Digital (21, 37, 12473)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12837, 12473, F128, 16) (dual of [12473, 12436, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12837, 16404, F128, 16) (dual of [16404, 16367, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(8) [i] based on
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-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(12837, 16404, F128, 16) (dual of [16404, 16367, 17]-code), using
(21, 37, large)-Net in Base 128 — Upper bound on s
There is no (21, 37, large)-net in base 128, because
- 14 times m-reduction [i] would yield (21, 23, large)-net in base 128, but