Best Known (30, 30+13, s)-Nets in Base 4
(30, 30+13, 240)-Net over F4 — Constructive and digital
Digital (30, 43, 240)-net over F4, using
- 41 times duplication [i] based on digital (29, 42, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 14, 80)-net over F64, using
(30, 30+13, 287)-Net over F4 — Digital
Digital (30, 43, 287)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(443, 287, F4, 13) (dual of [287, 244, 14]-code), using
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0) [i] based on linear OA(439, 265, F4, 13) (dual of [265, 226, 14]-code), using
- construction XX applied to C1 = C([79,89]), C2 = C([77,87]), C3 = C1 + C2 = C([79,87]), and C∩ = C1 ∩ C2 = C([77,89]) [i] based on
- linear OA(433, 255, F4, 11) (dual of [255, 222, 12]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {79,80,…,89}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(433, 255, F4, 11) (dual of [255, 222, 12]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {77,78,…,87}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(437, 255, F4, 13) (dual of [255, 218, 14]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {77,78,…,89}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(429, 255, F4, 9) (dual of [255, 226, 10]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {79,80,…,87}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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(41, 5, F4, 1) (dual of [5, 4, 2]-code) (see above)
- construction XX applied to C1 = C([79,89]), C2 = C([77,87]), C3 = C1 + C2 = C([79,87]), and C∩ = C1 ∩ C2 = C([77,89]) [i] based on
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0) [i] based on linear OA(439, 265, F4, 13) (dual of [265, 226, 14]-code), using
(30, 30+13, 16345)-Net in Base 4 — Upper bound on s
There is no (30, 43, 16346)-net in base 4, because
- 1 times m-reduction [i] would yield (30, 42, 16346)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 19 345554 417109 055878 810684 > 442 [i]