Best Known (53−12, 53, s)-Nets in Base 7
(53−12, 53, 2803)-Net over F7 — Constructive and digital
Digital (41, 53, 2803)-net over F7, using
- 71 times duplication [i] based on digital (40, 52, 2803)-net over F7, using
- net defined by OOA [i] based on linear OOA(752, 2803, F7, 12, 12) (dual of [(2803, 12), 33584, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 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(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(752, 16818, F7, 12) (dual of [16818, 16766, 13]-code), using
- net defined by OOA [i] based on linear OOA(752, 2803, F7, 12, 12) (dual of [(2803, 12), 33584, 13]-NRT-code), using
(53−12, 53, 16820)-Net over F7 — Digital
Digital (41, 53, 16820)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(753, 16820, F7, 12) (dual of [16820, 16767, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 12, F7, 1) (dual of [12, 11, 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
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
(53−12, 53, large)-Net in Base 7 — Upper bound on s
There is no (41, 53, large)-net in base 7, because
- 10 times m-reduction [i] would yield (41, 43, large)-net in base 7, but