Best Known (210, 243, s)-Nets in Base 4
(210, 243, 65536)-Net over F4 — Constructive and digital
Digital (210, 243, 65536)-net over F4, using
- 42 times duplication [i] based on digital (208, 241, 65536)-net over F4, using
- net defined by OOA [i] based on linear OOA(4241, 65536, F4, 33, 33) (dual of [(65536, 33), 2162447, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using
- net defined by OOA [i] based on linear OOA(4241, 65536, F4, 33, 33) (dual of [(65536, 33), 2162447, 34]-NRT-code), using
(210, 243, 349529)-Net over F4 — Digital
Digital (210, 243, 349529)-net over F4, using
- 41 times duplication [i] based on digital (209, 242, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4242, 349529, F4, 3, 33) (dual of [(349529, 3), 1048345, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4242, 1048587, F4, 33) (dual of [1048587, 1048345, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- OOA 3-folding [i] based on linear OA(4242, 1048587, F4, 33) (dual of [1048587, 1048345, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4242, 349529, F4, 3, 33) (dual of [(349529, 3), 1048345, 34]-NRT-code), using
(210, 243, large)-Net in Base 4 — Upper bound on s
There is no (210, 243, large)-net in base 4, because
- 31 times m-reduction [i] would yield (210, 212, large)-net in base 4, but