Best Known (80, 80+12, s)-Nets in Base 7
(80, 80+12, 1921603)-Net over F7 — Constructive and digital
Digital (80, 92, 1921603)-net over F7, using
- net defined by OOA [i] based on linear OOA(792, 1921603, F7, 14, 12) (dual of [(1921603, 14), 26902350, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(792, 5764810, F7, 2, 12) (dual of [(5764810, 2), 11529528, 13]-NRT-code), using
- trace code [i] based on linear OOA(4946, 2882405, F49, 2, 12) (dual of [(2882405, 2), 5764764, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4946, 5764810, F49, 12) (dual of [5764810, 5764764, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(4946, 5764810, F49, 12) (dual of [5764810, 5764764, 13]-code), using
- trace code [i] based on linear OOA(4946, 2882405, F49, 2, 12) (dual of [(2882405, 2), 5764764, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(792, 5764810, F7, 2, 12) (dual of [(5764810, 2), 11529528, 13]-NRT-code), using
(80, 80+12, large)-Net over F7 — Digital
Digital (80, 92, large)-net over F7, using
- 72 times duplication [i] based on digital (78, 90, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(790, large, F7, 12) (dual of [large, large−90, 13]-code), using
- trace code [i] based on linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- trace code [i] based on linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(790, large, F7, 12) (dual of [large, large−90, 13]-code), using
(80, 80+12, large)-Net in Base 7 — Upper bound on s
There is no (80, 92, large)-net in base 7, because
- 10 times m-reduction [i] would yield (80, 82, large)-net in base 7, but