Best Known (236−31, 236, s)-Nets in Base 4
(236−31, 236, 69907)-Net over F4 — Constructive and digital
Digital (205, 236, 69907)-net over F4, using
- net defined by OOA [i] based on linear OOA(4236, 69907, F4, 31, 31) (dual of [(69907, 31), 2166881, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4236, 1048606, F4, 31) (dual of [1048606, 1048370, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4236, 1048611, F4, 31) (dual of [1048611, 1048375, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 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(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4236, 1048611, F4, 31) (dual of [1048611, 1048375, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4236, 1048606, F4, 31) (dual of [1048606, 1048370, 32]-code), using
(236−31, 236, 404912)-Net over F4 — Digital
Digital (205, 236, 404912)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4236, 404912, F4, 2, 31) (dual of [(404912, 2), 809588, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4236, 524305, F4, 2, 31) (dual of [(524305, 2), 1048374, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4236, 1048610, F4, 31) (dual of [1048610, 1048374, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4236, 1048611, F4, 31) (dual of [1048611, 1048375, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 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(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4236, 1048611, F4, 31) (dual of [1048611, 1048375, 32]-code), using
- OOA 2-folding [i] based on linear OA(4236, 1048610, F4, 31) (dual of [1048610, 1048374, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(4236, 524305, F4, 2, 31) (dual of [(524305, 2), 1048374, 32]-NRT-code), using
(236−31, 236, large)-Net in Base 4 — Upper bound on s
There is no (205, 236, large)-net in base 4, because
- 29 times m-reduction [i] would yield (205, 207, large)-net in base 4, but