Best Known (25, 42, s)-Nets in Base 64
(25, 42, 592)-Net over F64 — Constructive and digital
Digital (25, 42, 592)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (16, 33, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
- digital (1, 9, 80)-net over F64, using
(25, 42, 2049)-Net in Base 64 — Constructive
(25, 42, 2049)-net in base 64, using
- base change [i] based on digital (19, 36, 2049)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 2049, F128, 17, 17) (dual of [(2049, 17), 34797, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12836, 16393, F128, 17) (dual of [16393, 16357, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [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(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12836, 16393, F128, 17) (dual of [16393, 16357, 18]-code), using
- net defined by OOA [i] based on linear OOA(12836, 2049, F128, 17, 17) (dual of [(2049, 17), 34797, 18]-NRT-code), using
(25, 42, 6031)-Net over F64 — Digital
Digital (25, 42, 6031)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6442, 6031, F64, 17) (dual of [6031, 5989, 18]-code), using
- 1924 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 18 times 0, 1, 69 times 0, 1, 208 times 0, 1, 531 times 0, 1, 1088 times 0) [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 1924 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 18 times 0, 1, 69 times 0, 1, 208 times 0, 1, 531 times 0, 1, 1088 times 0) [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
(25, 42, 6234)-Net in Base 64
(25, 42, 6234)-net in base 64, using
- base change [i] based on digital (19, 36, 6234)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 6234, F128, 2, 17) (dual of [(6234, 2), 12432, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 8198, F128, 2, 17) (dual of [(8198, 2), 16360, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [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(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(12836, 16396, F128, 17) (dual of [16396, 16360, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 8198, F128, 2, 17) (dual of [(8198, 2), 16360, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 6234, F128, 2, 17) (dual of [(6234, 2), 12432, 18]-NRT-code), using
(25, 42, large)-Net in Base 64 — Upper bound on s
There is no (25, 42, large)-net in base 64, because
- 15 times m-reduction [i] would yield (25, 27, large)-net in base 64, but