Best Known (223−40, 223, s)-Nets in Base 4
(223−40, 223, 1539)-Net over F4 — Constructive and digital
Digital (183, 223, 1539)-net over F4, using
- t-expansion [i] based on digital (182, 223, 1539)-net over F4, using
- 8 times m-reduction [i] based on digital (182, 231, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 77, 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, 77, 513)-net over F64, using
- 8 times m-reduction [i] based on digital (182, 231, 1539)-net over F4, using
(223−40, 223, 16433)-Net over F4 — Digital
Digital (183, 223, 16433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4223, 16433, F4, 40) (dual of [16433, 16210, 41]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4220, 16428, F4, 40) (dual of [16428, 16208, 41]-code), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(49, 44, F4, 5) (dual of [44, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- linear OA(4220, 16430, F4, 38) (dual of [16430, 16210, 39]-code), using Gilbert–Varšamov bound and bm = 4220 > Vbs−1(k−1) = 29858 617316 313925 591607 719692 708220 547096 826525 437527 811578 734306 021120 465026 309411 934242 775566 809740 637002 109302 382758 232533 569392 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 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(4220, 16428, F4, 40) (dual of [16428, 16208, 41]-code), using
- construction X with Varšamov bound [i] based on
(223−40, 223, large)-Net in Base 4 — Upper bound on s
There is no (183, 223, large)-net in base 4, because
- 38 times m-reduction [i] would yield (183, 185, large)-net in base 4, but