Best Known (52, 52+8, s)-Nets in Base 7
(52, 52+8, 2882404)-Net over F7 — Constructive and digital
Digital (52, 60, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(760, 2882404, F7, 10, 8) (dual of [(2882404, 10), 28823980, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(760, 5764809, F7, 2, 8) (dual of [(5764809, 2), 11529558, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(760, 5764810, F7, 2, 8) (dual of [(5764810, 2), 11529560, 9]-NRT-code), using
- trace code [i] based on linear OOA(4930, 2882405, F49, 2, 8) (dual of [(2882405, 2), 5764780, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4930, 5764810, F49, 8) (dual of [5764810, 5764780, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(7) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(4930, 5764810, F49, 8) (dual of [5764810, 5764780, 9]-code), using
- trace code [i] based on linear OOA(4930, 2882405, F49, 2, 8) (dual of [(2882405, 2), 5764780, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(760, 5764810, F7, 2, 8) (dual of [(5764810, 2), 11529560, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(760, 5764809, F7, 2, 8) (dual of [(5764809, 2), 11529558, 9]-NRT-code), using
(52, 52+8, large)-Net over F7 — Digital
Digital (52, 60, large)-net over F7, using
- 75 times duplication [i] based on digital (47, 55, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(755, large, F7, 8) (dual of [large, large−55, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(755, large, F7, 8) (dual of [large, large−55, 9]-code), using
(52, 52+8, large)-Net in Base 7 — Upper bound on s
There is no (52, 60, large)-net in base 7, because
- 6 times m-reduction [i] would yield (52, 54, large)-net in base 7, but