Best Known (209, 209+32, s)-Nets in Base 4
(209, 209+32, 65536)-Net over F4 — Constructive and digital
Digital (209, 241, 65536)-net over F4, using
- t-expansion [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
(209, 209+32, 356268)-Net over F4 — Digital
Digital (209, 241, 356268)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4241, 356268, F4, 2, 32) (dual of [(356268, 2), 712295, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4241, 524293, F4, 2, 32) (dual of [(524293, 2), 1048345, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4241, 1048586, F4, 32) (dual of [1048586, 1048345, 33]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(4242, 1048587, F4, 33) (dual of [1048587, 1048345, 34]-code), using
- OOA 2-folding [i] based on linear OA(4241, 1048586, F4, 32) (dual of [1048586, 1048345, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(4241, 524293, F4, 2, 32) (dual of [(524293, 2), 1048345, 33]-NRT-code), using
(209, 209+32, large)-Net in Base 4 — Upper bound on s
There is no (209, 241, large)-net in base 4, because
- 30 times m-reduction [i] would yield (209, 211, large)-net in base 4, but