Best Known (230−32, 230, s)-Nets in Base 4
(230−32, 230, 16388)-Net over F4 — Constructive and digital
Digital (198, 230, 16388)-net over F4, using
- net defined by OOA [i] based on linear OOA(4230, 16388, F4, 32, 32) (dual of [(16388, 32), 524186, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(4230, 262208, F4, 32) (dual of [262208, 261978, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4230, 262209, F4, 32) (dual of [262209, 261979, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4230, 262209, F4, 32) (dual of [262209, 261979, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(4230, 262208, F4, 32) (dual of [262208, 261978, 33]-code), using
(230−32, 230, 158244)-Net over F4 — Digital
Digital (198, 230, 158244)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4230, 158244, F4, 32) (dual of [158244, 158014, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4230, 262209, F4, 32) (dual of [262209, 261979, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4217, 262145, F4, 33) (dual of [262145, 261928, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4230, 262209, F4, 32) (dual of [262209, 261979, 33]-code), using
(230−32, 230, large)-Net in Base 4 — Upper bound on s
There is no (198, 230, large)-net in base 4, because
- 30 times m-reduction [i] would yield (198, 200, large)-net in base 4, but