Best Known (213, 213+38, s)-Nets in Base 4
(213, 213+38, 3470)-Net over F4 — Constructive and digital
Digital (213, 251, 3470)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (187, 225, 3449)-net over F4, using
- net defined by OOA [i] based on linear OOA(4225, 3449, F4, 38, 38) (dual of [(3449, 38), 130837, 39]-NRT-code), using
- OA 19-folding and stacking [i] based on linear OA(4225, 65531, F4, 38) (dual of [65531, 65306, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using
- an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- discarding factors / shortening the dual code based on linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using
- OA 19-folding and stacking [i] based on linear OA(4225, 65531, F4, 38) (dual of [65531, 65306, 39]-code), using
- net defined by OOA [i] based on linear OOA(4225, 3449, F4, 38, 38) (dual of [(3449, 38), 130837, 39]-NRT-code), using
- digital (7, 26, 21)-net over F4, using
(213, 213+38, 65628)-Net over F4 — Digital
Digital (213, 251, 65628)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4251, 65628, F4, 38) (dual of [65628, 65377, 39]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4249, 65624, F4, 38) (dual of [65624, 65375, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(26) [i] based on
- linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(424, 88, F4, 10) (dual of [88, 64, 11]-code), using
- construction X applied to Ce(37) ⊂ Ce(26) [i] based on
- linear OA(4249, 65626, F4, 37) (dual of [65626, 65377, 38]-code), using Gilbert–Varšamov bound and bm = 4249 > Vbs−1(k−1) = 103808 973300 366592 314332 094137 619346 086037 747598 309036 024469 684879 650426 039868 877548 919250 796491 391265 053624 899410 906359 079751 729178 475217 563193 392876 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(4249, 65624, F4, 38) (dual of [65624, 65375, 39]-code), using
- construction X with Varšamov bound [i] based on
(213, 213+38, large)-Net in Base 4 — Upper bound on s
There is no (213, 251, large)-net in base 4, because
- 36 times m-reduction [i] would yield (213, 215, large)-net in base 4, but