Best Known (55, 55+16, s)-Nets in Base 7
(55, 55+16, 2103)-Net over F7 — Constructive and digital
Digital (55, 71, 2103)-net over F7, using
- 71 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(770, 2103, F7, 16, 16) (dual of [(2103, 16), 33578, 17]-NRT-code), using
(55, 55+16, 16828)-Net over F7 — Digital
Digital (55, 71, 16828)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 16828, F7, 16) (dual of [16828, 16757, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [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(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(74, 20, F7, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,7)), 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(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
(55, 55+16, large)-Net in Base 7 — Upper bound on s
There is no (55, 71, large)-net in base 7, because
- 14 times m-reduction [i] would yield (55, 57, large)-net in base 7, but