Best Known (231−29, 231, s)-Nets in Base 4
(231−29, 231, 74916)-Net over F4 — Constructive and digital
Digital (202, 231, 74916)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (183, 212, 74899)-net over F4, using
- net defined by OOA [i] based on linear OOA(4212, 74899, F4, 29, 29) (dual of [(74899, 29), 2171859, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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(28) ⊂ Ce(26) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- net defined by OOA [i] based on linear OOA(4212, 74899, F4, 29, 29) (dual of [(74899, 29), 2171859, 30]-NRT-code), using
- digital (5, 19, 17)-net over F4, using
(231−29, 231, 524329)-Net over F4 — Digital
Digital (202, 231, 524329)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4231, 524329, F4, 2, 29) (dual of [(524329, 2), 1048427, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4231, 1048658, F4, 29) (dual of [1048658, 1048427, 30]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4226, 1048651, F4, 29) (dual of [1048651, 1048425, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(415, 75, F4, 7) (dual of [75, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4226, 1048653, F4, 25) (dual of [1048653, 1048427, 26]-code), using Gilbert–Varšamov bound and bm = 4226 > Vbs−1(k−1) = 1 423137 291011 356040 720721 505956 111213 655175 185098 411660 547843 175209 608718 914415 992969 155079 476181 000695 286357 906932 832081 801487 820596 [i]
- linear OA(43, 5, F4, 3) (dual of [5, 2, 4]-code or 5-arc in PG(2,4) or 5-cap in PG(2,4)), using
- extended Reed–Solomon code RSe(2,4) [i]
- Simplex code S(2,4) [i]
- linear OA(4226, 1048651, F4, 29) (dual of [1048651, 1048425, 30]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(4231, 1048658, F4, 29) (dual of [1048658, 1048427, 30]-code), using
(231−29, 231, large)-Net in Base 4 — Upper bound on s
There is no (202, 231, large)-net in base 4, because
- 27 times m-reduction [i] would yield (202, 204, large)-net in base 4, but