Best Known (51−9, 51, s)-Nets in Base 8
(51−9, 51, 524289)-Net over F8 — Constructive and digital
Digital (42, 51, 524289)-net over F8, using
- net defined by OOA [i] based on linear OOA(851, 524289, F8, 9, 9) (dual of [(524289, 9), 4718550, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(851, 2097157, F8, 9) (dual of [2097157, 2097106, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(851, 2097160, F8, 9) (dual of [2097160, 2097109, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(843, 2097152, F8, 7) (dual of [2097152, 2097109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 8, F8, 1) (dual of [8, 7, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(851, 2097160, F8, 9) (dual of [2097160, 2097109, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(851, 2097157, F8, 9) (dual of [2097157, 2097106, 10]-code), using
(51−9, 51, 1362895)-Net over F8 — Digital
Digital (42, 51, 1362895)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(851, 1362895, F8, 9) (dual of [1362895, 1362844, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(851, 2097160, F8, 9) (dual of [2097160, 2097109, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(843, 2097152, F8, 7) (dual of [2097152, 2097109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 8, F8, 1) (dual of [8, 7, 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(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(851, 2097160, F8, 9) (dual of [2097160, 2097109, 10]-code), using
(51−9, 51, large)-Net in Base 8 — Upper bound on s
There is no (42, 51, large)-net in base 8, because
- 7 times m-reduction [i] would yield (42, 44, large)-net in base 8, but