Best Known (174, 174+31, s)-Nets in Base 4
(174, 174+31, 4386)-Net over F4 — Constructive and digital
Digital (174, 205, 4386)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 20, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (154, 185, 4369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using
- net defined by OOA [i] based on linear OOA(4185, 4369, F4, 31, 31) (dual of [(4369, 31), 135254, 32]-NRT-code), using
- digital (5, 20, 17)-net over F4, using
(174, 174+31, 65613)-Net over F4 — Digital
Digital (174, 205, 65613)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4205, 65613, F4, 31) (dual of [65613, 65408, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4204, 65611, F4, 31) (dual of [65611, 65407, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(419, 75, F4, 8) (dual of [75, 56, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 87, F4, 8) (dual of [87, 68, 9]-code), using
- construction X applied to Ce(30) ⊂ Ce(21) [i] based on
- linear OA(4204, 65612, F4, 30) (dual of [65612, 65408, 31]-code), using Gilbert–Varšamov bound and bm = 4204 > Vbs−1(k−1) = 379 906601 987730 540092 892364 921857 359104 273444 494271 037689 623164 450537 337911 911747 580424 023009 755976 925312 815850 604907 890912 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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(4204, 65611, F4, 31) (dual of [65611, 65407, 32]-code), using
- construction X with Varšamov bound [i] based on
(174, 174+31, large)-Net in Base 4 — Upper bound on s
There is no (174, 205, large)-net in base 4, because
- 29 times m-reduction [i] would yield (174, 176, large)-net in base 4, but