Best Known (206, 206+27, s)-Nets in Base 4
(206, 206+27, 322643)-Net over F4 — Constructive and digital
Digital (206, 233, 322643)-net over F4, using
- 42 times duplication [i] based on digital (204, 231, 322643)-net over F4, using
- net defined by OOA [i] based on linear OOA(4231, 322643, F4, 27, 27) (dual of [(322643, 27), 8711130, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4231, 4194360, F4, 27) (dual of [4194360, 4194129, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 4194368, F4, 27) (dual of [4194368, 4194137, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4231, 4194368, F4, 27) (dual of [4194368, 4194137, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4231, 4194360, F4, 27) (dual of [4194360, 4194129, 28]-code), using
- net defined by OOA [i] based on linear OOA(4231, 322643, F4, 27, 27) (dual of [(322643, 27), 8711130, 28]-NRT-code), using
(206, 206+27, 2037346)-Net over F4 — Digital
Digital (206, 233, 2037346)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4233, 2037346, F4, 2, 27) (dual of [(2037346, 2), 4074459, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4233, 2097185, F4, 2, 27) (dual of [(2097185, 2), 4194137, 28]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4232, 2097185, F4, 2, 27) (dual of [(2097185, 2), 4194138, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4232, 4194370, F4, 27) (dual of [4194370, 4194138, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(411, 66, F4, 5) (dual of [66, 55, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(47, 64, F4, 3) (dual of [64, 57, 4]-code or 64-cap in PG(6,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 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(4) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(411, 68, F4, 5) (dual of [68, 57, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(4232, 4194370, F4, 27) (dual of [4194370, 4194138, 28]-code), using
- 41 times duplication [i] based on linear OOA(4232, 2097185, F4, 2, 27) (dual of [(2097185, 2), 4194138, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4233, 2097185, F4, 2, 27) (dual of [(2097185, 2), 4194137, 28]-NRT-code), using
(206, 206+27, large)-Net in Base 4 — Upper bound on s
There is no (206, 233, large)-net in base 4, because
- 25 times m-reduction [i] would yield (206, 208, large)-net in base 4, but