Best Known (220−42, 220, s)-Nets in Base 4
(220−42, 220, 1539)-Net over F4 — Constructive and digital
Digital (178, 220, 1539)-net over F4, using
- 5 times m-reduction [i] based on digital (178, 225, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
(220−42, 220, 10366)-Net over F4 — Digital
Digital (178, 220, 10366)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 10366, F4, 42) (dual of [10366, 10146, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4220, 16394, F4, 42) (dual of [16394, 16174, 43]-code), using
- construction XX applied to Ce(41) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- 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(4204, 16384, F4, 39) (dual of [16384, 16180, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(41) ⊂ Ce(40) ⊂ Ce(38) [i] based on
- discarding factors / shortening the dual code based on linear OA(4220, 16394, F4, 42) (dual of [16394, 16174, 43]-code), using
(220−42, 220, 5870366)-Net in Base 4 — Upper bound on s
There is no (178, 220, 5870367)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2 839220 765375 007458 224824 152710 068985 973235 252677 592431 338732 324913 490190 137807 473818 699248 867737 273104 853684 097982 125821 633183 889718 > 4220 [i]