Best Known (54, 54+17, s)-Nets in Base 7
(54, 54+17, 2101)-Net over F7 — Constructive and digital
Digital (54, 71, 2101)-net over F7, using
- net defined by OOA [i] based on linear OOA(771, 2101, F7, 17, 17) (dual of [(2101, 17), 35646, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(771, 16809, F7, 17) (dual of [16809, 16738, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 16812, F7, 17) (dual of [16812, 16741, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(771, 16812, F7, 17) (dual of [16812, 16741, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(771, 16809, F7, 17) (dual of [16809, 16738, 18]-code), using
(54, 54+17, 9397)-Net over F7 — Digital
Digital (54, 71, 9397)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 9397, F7, 17) (dual of [9397, 9326, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using
(54, 54+17, large)-Net in Base 7 — Upper bound on s
There is no (54, 71, large)-net in base 7, because
- 15 times m-reduction [i] would yield (54, 56, large)-net in base 7, but