Best Known (57, 57+16, s)-Nets in Base 8
(57, 57+16, 4097)-Net over F8 — Constructive and digital
Digital (57, 73, 4097)-net over F8, using
- t-expansion [i] based on digital (56, 73, 4097)-net over F8, using
- net defined by OOA [i] based on linear OOA(873, 4097, F8, 17, 17) (dual of [(4097, 17), 69576, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(873, 32777, F8, 17) (dual of [32777, 32704, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(873, 32777, F8, 17) (dual of [32777, 32704, 18]-code), using
- net defined by OOA [i] based on linear OOA(873, 4097, F8, 17, 17) (dual of [(4097, 17), 69576, 18]-NRT-code), using
(57, 57+16, 32781)-Net over F8 — Digital
Digital (57, 73, 32781)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(873, 32781, F8, 16) (dual of [32781, 32708, 17]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(856, 32768, F8, 13) (dual of [32768, 32712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
(57, 57+16, large)-Net in Base 8 — Upper bound on s
There is no (57, 73, large)-net in base 8, because
- 14 times m-reduction [i] would yield (57, 59, large)-net in base 8, but