Best Known (207−31, 207, s)-Nets in Base 4
(207−31, 207, 4390)-Net over F4 — Constructive and digital
Digital (176, 207, 4390)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (154, 185, 4369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- digital (7, 22, 21)-net over F4, using
(207−31, 207, 65623)-Net over F4 — Digital
Digital (176, 207, 65623)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4207, 65623, F4, 31) (dual of [65623, 65416, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4206, 65621, F4, 31) (dual of [65621, 65415, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(20) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(421, 85, F4, 9) (dual of [85, 64, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(421, 86, F4, 9) (dual of [86, 65, 10]-code), using
- construction X applied to Ce(30) ⊂ Ce(20) [i] based on
- linear OA(4206, 65622, F4, 30) (dual of [65622, 65416, 31]-code), using Gilbert–Varšamov bound and bm = 4206 > Vbs−1(k−1) = 381 589724 883936 498023 297673 565779 660837 181532 080839 077730 990767 589530 616414 333849 409116 494121 178902 114688 283165 970092 699392 [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(4206, 65621, F4, 31) (dual of [65621, 65415, 32]-code), using
- construction X with Varšamov bound [i] based on
(207−31, 207, large)-Net in Base 4 — Upper bound on s
There is no (176, 207, large)-net in base 4, because
- 29 times m-reduction [i] would yield (176, 178, large)-net in base 4, but