Best Known (194−31, 194, s)-Nets in Base 4
(194−31, 194, 4371)-Net over F4 — Constructive and digital
Digital (163, 194, 4371)-net over F4, using
- 43 times duplication [i] based on digital (160, 191, 4371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 4371, F4, 31, 31) (dual of [(4371, 31), 135310, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4191, 65566, F4, 31) (dual of [65566, 65375, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 65567, F4, 31) (dual of [65567, 65376, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 65567, F4, 31) (dual of [65567, 65376, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4191, 65566, F4, 31) (dual of [65566, 65375, 32]-code), using
- net defined by OOA [i] based on linear OOA(4191, 4371, F4, 31, 31) (dual of [(4371, 31), 135310, 32]-NRT-code), using
(194−31, 194, 39497)-Net over F4 — Digital
Digital (163, 194, 39497)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4194, 39497, F4, 31) (dual of [39497, 39303, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 65554, F4, 31) (dual of [65554, 65360, 32]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4177, 65537, F4, 29) (dual of [65537, 65360, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4194, 65554, F4, 31) (dual of [65554, 65360, 32]-code), using
(194−31, 194, large)-Net in Base 4 — Upper bound on s
There is no (163, 194, large)-net in base 4, because
- 29 times m-reduction [i] would yield (163, 165, large)-net in base 4, but