Best Known (54, 54+16, s)-Nets in Base 7
(54, 54+16, 2103)-Net over F7 — Constructive and digital
Digital (54, 70, 2103)-net over F7, using
- net defined by OOA [i] based on linear OOA(770, 2103, F7, 16, 16) (dual of [(2103, 16), 33578, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(770, 16824, F7, 16) (dual of [16824, 16754, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(770, 16826, F7, 16) (dual of [16826, 16756, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(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(74, 19, F7, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,7)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(770, 16826, F7, 16) (dual of [16826, 16756, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(770, 16824, F7, 16) (dual of [16824, 16754, 17]-code), using
(54, 54+16, 14729)-Net over F7 — Digital
Digital (54, 70, 14729)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(770, 14729, F7, 16) (dual of [14729, 14659, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(770, 16826, F7, 16) (dual of [16826, 16756, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(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(74, 19, F7, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,7)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(770, 16826, F7, 16) (dual of [16826, 16756, 17]-code), using
(54, 54+16, large)-Net in Base 7 — Upper bound on s
There is no (54, 70, large)-net in base 7, because
- 14 times m-reduction [i] would yield (54, 56, large)-net in base 7, but