Best Known (155, 155+25, s)-Nets in Base 4
(155, 155+25, 21862)-Net over F4 — Constructive and digital
Digital (155, 180, 21862)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 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 (138, 163, 21845)-net over F4, using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- digital (5, 17, 17)-net over F4, using
(155, 155+25, 152363)-Net over F4 — Digital
Digital (155, 180, 152363)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4180, 152363, F4, 25) (dual of [152363, 152183, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, 262213, F4, 25) (dual of [262213, 262033, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [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 C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4180, 262213, F4, 25) (dual of [262213, 262033, 26]-code), using
(155, 155+25, large)-Net in Base 4 — Upper bound on s
There is no (155, 180, large)-net in base 4, because
- 23 times m-reduction [i] would yield (155, 157, large)-net in base 4, but