Best Known (43, 43+9, s)-Nets in Base 7
(43, 43+9, 205888)-Net over F7 — Constructive and digital
Digital (43, 52, 205888)-net over F7, using
- net defined by OOA [i] based on linear OOA(752, 205888, F7, 9, 9) (dual of [(205888, 9), 1852940, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(752, 823553, F7, 9) (dual of [823553, 823501, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(752, 823553, F7, 9) (dual of [823553, 823501, 10]-code), using
(43, 43+9, 808925)-Net over F7 — Digital
Digital (43, 52, 808925)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 808925, F7, 9) (dual of [808925, 808873, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(752, 823553, F7, 9) (dual of [823553, 823501, 10]-code), using
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 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(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(8) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(752, 823553, F7, 9) (dual of [823553, 823501, 10]-code), using
(43, 43+9, large)-Net in Base 7 — Upper bound on s
There is no (43, 52, large)-net in base 7, because
- 7 times m-reduction [i] would yield (43, 45, large)-net in base 7, but