Best Known (153, 185, s)-Nets in Base 4
(153, 185, 1539)-Net over F4 — Constructive and digital
Digital (153, 185, 1539)-net over F4, using
- t-expansion [i] based on digital (152, 185, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (152, 186, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 62, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 62, 513)-net over F64, using
- 1 times m-reduction [i] based on digital (152, 186, 1539)-net over F4, using
(153, 185, 16446)-Net over F4 — Digital
Digital (153, 185, 16446)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4185, 16446, F4, 32) (dual of [16446, 16261, 33]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4182, 16440, F4, 32) (dual of [16440, 16258, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4182, 16443, F4, 31) (dual of [16443, 16261, 32]-code), using Gilbert–Varšamov bound and bm = 4182 > Vbs−1(k−1) = 2 277113 684771 857228 524908 643950 504284 563837 243592 415876 870514 007819 190424 282275 896976 604567 345972 185866 866544 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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(4182, 16440, F4, 32) (dual of [16440, 16258, 33]-code), using
- construction X with Varšamov bound [i] based on
(153, 185, large)-Net in Base 4 — Upper bound on s
There is no (153, 185, large)-net in base 4, because
- 30 times m-reduction [i] would yield (153, 155, large)-net in base 4, but