Best Known (110−16, 110, s)-Nets in Base 4
(110−16, 110, 32770)-Net over F4 — Constructive and digital
Digital (94, 110, 32770)-net over F4, using
- net defined by OOA [i] based on linear OOA(4110, 32770, F4, 16, 16) (dual of [(32770, 16), 524210, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4110, 262160, F4, 16) (dual of [262160, 262050, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 262163, F4, 16) (dual of [262163, 262053, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4110, 262163, F4, 16) (dual of [262163, 262053, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(4110, 262160, F4, 16) (dual of [262160, 262050, 17]-code), using
(110−16, 110, 131082)-Net over F4 — Digital
Digital (94, 110, 131082)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4110, 131082, F4, 2, 16) (dual of [(131082, 2), 262054, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4110, 262164, F4, 16) (dual of [262164, 262054, 17]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(4110, 262164, F4, 16) (dual of [262164, 262054, 17]-code), using
(110−16, 110, large)-Net in Base 4 — Upper bound on s
There is no (94, 110, large)-net in base 4, because
- 14 times m-reduction [i] would yield (94, 96, large)-net in base 4, but