Best Known (36−11, 36, s)-Nets in Base 4
(36−11, 36, 240)-Net over F4 — Constructive and digital
Digital (25, 36, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 12, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
(36−11, 36, 276)-Net over F4 — Digital
Digital (25, 36, 276)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(436, 276, F4, 11) (dual of [276, 240, 12]-code), using
- 10 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0) [i] based on linear OA(433, 263, F4, 11) (dual of [263, 230, 12]-code), using
- construction XX applied to C1 = C([254,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([254,9]) [i] based on
- linear OA(429, 255, F4, 10) (dual of [255, 226, 11]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(429, 255, F4, 10) (dual of [255, 226, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(433, 255, F4, 11) (dual of [255, 222, 12]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(425, 255, F4, 9) (dual of [255, 230, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 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
- linear OA(40, 4, F4, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([254,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([254,9]) [i] based on
- 10 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0) [i] based on linear OA(433, 263, F4, 11) (dual of [263, 230, 12]-code), using
(36−11, 36, 14224)-Net in Base 4 — Upper bound on s
There is no (25, 36, 14225)-net in base 4, because
- 1 times m-reduction [i] would yield (25, 35, 14225)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1180 989389 586469 258336 > 435 [i]