Best Known (46, 57, s)-Nets in Base 8
(46, 57, 52431)-Net over F8 — Constructive and digital
Digital (46, 57, 52431)-net over F8, using
- 81 times duplication [i] based on digital (45, 56, 52431)-net over F8, using
- net defined by OOA [i] based on linear OOA(856, 52431, F8, 11, 11) (dual of [(52431, 11), 576685, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(856, 262156, F8, 11) (dual of [262156, 262100, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(856, 262157, F8, 11) (dual of [262157, 262101, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(856, 262156, F8, 11) (dual of [262156, 262100, 12]-code), using
- net defined by OOA [i] based on linear OOA(856, 52431, F8, 11, 11) (dual of [(52431, 11), 576685, 12]-NRT-code), using
(46, 57, 246531)-Net over F8 — Digital
Digital (46, 57, 246531)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 246531, F8, 11) (dual of [246531, 246474, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 262159, F8, 11) (dual of [262159, 262102, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(856, 262158, F8, 11) (dual of [262158, 262102, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(813, 14, F8, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,8)), using
- dual of repetition code with length 14 [i]
- linear OA(81, 14, F8, 1) (dual of [14, 13, 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
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(856, 262158, F8, 11) (dual of [262158, 262102, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 262159, F8, 11) (dual of [262159, 262102, 12]-code), using
(46, 57, large)-Net in Base 8 — Upper bound on s
There is no (46, 57, large)-net in base 8, because
- 9 times m-reduction [i] would yield (46, 48, large)-net in base 8, but