Best Known (149, 149+23, s)-Nets in Base 4
(149, 149+23, 95326)-Net over F4 — Constructive and digital
Digital (149, 172, 95326)-net over F4, using
- net defined by OOA [i] based on linear OOA(4172, 95326, F4, 23, 23) (dual of [(95326, 23), 2192326, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4172, 1048587, F4, 23) (dual of [1048587, 1048415, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 1048597, F4, 23) (dual of [1048597, 1048425, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 1048597, F4, 23) (dual of [1048597, 1048425, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4172, 1048587, F4, 23) (dual of [1048587, 1048415, 24]-code), using
(149, 149+23, 362807)-Net over F4 — Digital
Digital (149, 172, 362807)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4172, 362807, F4, 2, 23) (dual of [(362807, 2), 725442, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4172, 524298, F4, 2, 23) (dual of [(524298, 2), 1048424, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4172, 1048596, F4, 23) (dual of [1048596, 1048424, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 1048597, F4, 23) (dual of [1048597, 1048425, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 1048597, F4, 23) (dual of [1048597, 1048425, 24]-code), using
- OOA 2-folding [i] based on linear OA(4172, 1048596, F4, 23) (dual of [1048596, 1048424, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(4172, 524298, F4, 2, 23) (dual of [(524298, 2), 1048424, 24]-NRT-code), using
(149, 149+23, large)-Net in Base 4 — Upper bound on s
There is no (149, 172, large)-net in base 4, because
- 21 times m-reduction [i] would yield (149, 151, large)-net in base 4, but