Best Known (171, 171+37, s)-Nets in Base 4
(171, 171+37, 1539)-Net over F4 — Constructive and digital
Digital (171, 208, 1539)-net over F4, using
- t-expansion [i] based on digital (170, 208, 1539)-net over F4, using
- 5 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 71, 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, 71, 513)-net over F64, using
- 5 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
(171, 171+37, 16447)-Net over F4 — Digital
Digital (171, 208, 16447)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4208, 16447, F4, 37) (dual of [16447, 16239, 38]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4204, 16440, F4, 37) (dual of [16440, 16236, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4204, 16443, F4, 35) (dual of [16443, 16239, 36]-code), using Gilbert–Varšamov bound and bm = 4204 > Vbs−1(k−1) = 12 019539 494527 898140 633992 342083 572364 819766 679581 739340 328200 830035 120618 111899 539454 827818 927825 806185 000388 754046 660564 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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(4204, 16440, F4, 37) (dual of [16440, 16236, 38]-code), using
- construction X with Varšamov bound [i] based on
(171, 171+37, large)-Net in Base 4 — Upper bound on s
There is no (171, 208, large)-net in base 4, because
- 35 times m-reduction [i] would yield (171, 173, large)-net in base 4, but