Best Known (37, 48, s)-Nets in Base 4
(37, 48, 1032)-Net over F4 — Constructive and digital
Digital (37, 48, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 12, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
(37, 48, 1248)-Net over F4 — Digital
Digital (37, 48, 1248)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(448, 1248, F4, 11) (dual of [1248, 1200, 12]-code), using
- 208 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 15 times 0, 1, 31 times 0, 1, 56 times 0, 1, 92 times 0) [i] based on linear OA(441, 1033, F4, 11) (dual of [1033, 992, 12]-code), using
- construction XX applied to C1 = C([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [i] based on
- linear OA(436, 1023, F4, 10) (dual of [1023, 987, 11]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(436, 1023, F4, 10) (dual of [1023, 987, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(441, 1023, F4, 11) (dual of [1023, 982, 12]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(431, 1023, F4, 9) (dual of [1023, 992, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [i] based on
- 208 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 15 times 0, 1, 31 times 0, 1, 56 times 0, 1, 92 times 0) [i] based on linear OA(441, 1033, F4, 11) (dual of [1033, 992, 12]-code), using
(37, 48, 396346)-Net in Base 4 — Upper bound on s
There is no (37, 48, 396347)-net in base 4, because
- 1 times m-reduction [i] would yield (37, 47, 396347)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 19807 195701 367506 043486 346116 > 447 [i]