Best Known (91, 91+16, s)-Nets in Base 4
(91, 91+16, 8206)-Net over F4 — Constructive and digital
Digital (91, 107, 8206)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (80, 96, 8192)-net over F4, using
- net defined by OOA [i] based on linear OOA(496, 8192, F4, 16, 16) (dual of [(8192, 16), 130976, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(496, 65536, F4, 16) (dual of [65536, 65440, 17]-code), using
- 1 times truncation [i] based on linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(496, 65536, F4, 16) (dual of [65536, 65440, 17]-code), using
- net defined by OOA [i] based on linear OOA(496, 8192, F4, 16, 16) (dual of [(8192, 16), 130976, 17]-NRT-code), using
- digital (3, 11, 14)-net over F4, using
(91, 91+16, 65587)-Net over F4 — Digital
Digital (91, 107, 65587)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4107, 65587, F4, 16) (dual of [65587, 65480, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4106, 65585, F4, 16) (dual of [65585, 65479, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(49, 49, F4, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(4106, 65586, F4, 15) (dual of [65586, 65480, 16]-code), using Gilbert–Varšamov bound and bm = 4106 > Vbs−1(k−1) = 1492 728460 061535 490508 264951 656682 458478 861342 597163 103179 070036 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4106, 65585, F4, 16) (dual of [65585, 65479, 17]-code), using
- construction X with Varšamov bound [i] based on
(91, 91+16, large)-Net in Base 4 — Upper bound on s
There is no (91, 107, large)-net in base 4, because
- 14 times m-reduction [i] would yield (91, 93, large)-net in base 4, but