Best Known (194, 194+31, s)-Nets in Base 4
(194, 194+31, 17486)-Net over F4 — Constructive and digital
Digital (194, 225, 17486)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (177, 208, 17476)-net over F4, using
- net defined by OOA [i] based on linear OOA(4208, 17476, F4, 31, 31) (dual of [(17476, 31), 541548, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4208, 262141, F4, 31) (dual of [262141, 261933, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4208, 262141, F4, 31) (dual of [262141, 261933, 32]-code), using
- net defined by OOA [i] based on linear OOA(4208, 17476, F4, 31, 31) (dual of [(17476, 31), 541548, 32]-NRT-code), using
- digital (2, 17, 10)-net over F4, using
(194, 194+31, 173915)-Net over F4 — Digital
Digital (194, 225, 173915)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4225, 173915, F4, 31) (dual of [173915, 173690, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4225, 262212, F4, 31) (dual of [262212, 261987, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(417, 68, F4, 7) (dual of [68, 51, 8]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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
- linear OA(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- linear OA(49, 17, F4, 7) (dual of [17, 8, 8]-code), using
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4225, 262212, F4, 31) (dual of [262212, 261987, 32]-code), using
(194, 194+31, large)-Net in Base 4 — Upper bound on s
There is no (194, 225, large)-net in base 4, because
- 29 times m-reduction [i] would yield (194, 196, large)-net in base 4, but