Best Known (132−18, 132, s)-Nets in Base 4
(132−18, 132, 116509)-Net over F4 — Constructive and digital
Digital (114, 132, 116509)-net over F4, using
- 41 times duplication [i] based on digital (113, 131, 116509)-net over F4, using
- net defined by OOA [i] based on linear OOA(4131, 116509, F4, 18, 18) (dual of [(116509, 18), 2097031, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4131, 1048581, F4, 18) (dual of [1048581, 1048450, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4131, 1048581, F4, 18) (dual of [1048581, 1048450, 19]-code), using
- net defined by OOA [i] based on linear OOA(4131, 116509, F4, 18, 18) (dual of [(116509, 18), 2097031, 19]-NRT-code), using
(132−18, 132, 353577)-Net over F4 — Digital
Digital (114, 132, 353577)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4132, 353577, F4, 2, 18) (dual of [(353577, 2), 707022, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4132, 524293, F4, 2, 18) (dual of [(524293, 2), 1048454, 19]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4131, 524293, F4, 2, 18) (dual of [(524293, 2), 1048455, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4131, 1048586, F4, 18) (dual of [1048586, 1048455, 19]-code), using
- 41 times duplication [i] based on linear OOA(4131, 524293, F4, 2, 18) (dual of [(524293, 2), 1048455, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4132, 524293, F4, 2, 18) (dual of [(524293, 2), 1048454, 19]-NRT-code), using
(132−18, 132, large)-Net in Base 4 — Upper bound on s
There is no (114, 132, large)-net in base 4, because
- 16 times m-reduction [i] would yield (114, 116, large)-net in base 4, but