Best Known (36, 36+8, s)-Nets in Base 7
(36, 36+8, 205888)-Net over F7 — Constructive and digital
Digital (36, 44, 205888)-net over F7, using
- net defined by OOA [i] based on linear OOA(744, 205888, F7, 8, 8) (dual of [(205888, 8), 1647060, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(744, 823552, F7, 8) (dual of [823552, 823508, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(78, 9, F7, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,7)), using
- dual of repetition code with length 9 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(744, 823552, F7, 8) (dual of [823552, 823508, 9]-code), using
(36, 36+8, 568335)-Net over F7 — Digital
Digital (36, 44, 568335)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(744, 568335, F7, 8) (dual of [568335, 568291, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(744, 823551, F7, 8) (dual of [823551, 823507, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(736, 823543, F7, 6) (dual of [823543, 823507, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 8, F7, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(744, 823551, F7, 8) (dual of [823551, 823507, 9]-code), using
(36, 36+8, large)-Net in Base 7 — Upper bound on s
There is no (36, 44, large)-net in base 7, because
- 6 times m-reduction [i] would yield (36, 38, large)-net in base 7, but