1 Introduction
2 Preliminaries and known partial classification
\(t=4\) | \(t=5\) | \(t=6\) | \(t=7\) | |
---|---|---|---|---|
\((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | |
\(\mathbb {Z}_4\) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |
(2, 1) | (2, 2) | (2, 3) | (2, 4) | |
\(\mathbb {Z}_{2^3}\) | (1, 0, 2) | (1, 0, 3) | (1, 0, 4) | (1, 0, 5) |
(1, 1, 0) | (1, 1, 1) | (1, 1, 2) | (1, 1, 3) | |
\(\mathbb {Z}_{2^4}\) | (1, 0, 0, 1) | (1, 0, 0, 2) | (1, 0, 0, 3) | (1, 0, 0, 4) |
(1, 0, 1, 0) | (1, 0, 1, 1) | (1, 0, 1, 2) | ||
\(\mathbb {Z}_{2^5}\) | (1, 0, 0, 0, 0) | (1, 0, 0, 0, 1) | (1, 0, 0, 0, 2) | (1, 0, 0, 0, 3) |
(1, 0, 0, 1, 0) | (1, 0, 0, 1, 1) | |||
\(\mathbb {Z}_{2^6}\) | (1, 0, 0, 0, 0, 0) | (1, 0, 0, 0, 0, 1) | (1, 0, 0, 0, 0, 2) | |
(1, 0, 0, 0, 1, 0) | ||||
\(\mathbb {Z}_{2^7}\) | (1, 0, 0, 0, 0, 0, 0) | (1, 0, 0, 0, 0, 0, 1) | ||
\(\mathbb {Z}_{2^8}\) | (1, 0, 0, 0, 0, 0, 0, 0) |
\(t=4\) | \(t=5\) | \(t=6\) | \(t=7\) | |
---|---|---|---|---|
\((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | \((t_1,\ldots ,t_s)\) | |
\(\mathbb {Z}_{p^2}\) | (1, 3) | (1, 4) | (1, 5) | (1, 6) |
\(\mathbb {Z}_{p^3}\) | (1, 0, 2) | (1, 0, 3) | (1, 0, 4) | (1, 0, 5) |
\(\mathbb {Z}_{p^4}\) | (1, 0, 0, 1) | (1, 0, 0, 2) | (1, 0, 0, 3) | (1, 0, 0, 4) |
\(\mathbb {Z}_{p^5}\) | (1, 0, 0, 0, 0) | (1, 0, 0, 0, 1) | (1, 0, 0, 0, 2) | (1, 0, 0, 0, 3) |
\(\mathbb {Z}_{p^6}\) | (1, 0, 0, 0, 0, 0) | (1, 0, 0, 0, 0, 1) | (1, 0, 0, 0, 0, 2) | |
\(\mathbb {Z}_{p^7}\) | (1, 0, 0, 0, 0, 0, 0) | (1, 0, 0, 0, 0, 0, 1) | ||
\(\mathbb {Z}_{p^8}\) | (1, 0, 0, 0, 0, 0, 0, 0) |
\(t=4\) | \(t=5\) | \(t=6\) | \(t=7\) | |||||
---|---|---|---|---|---|---|---|---|
\((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | |
\(\mathbb {Z}_9\) | (2, 1) | (6,3) | (2, 2) | (7,4) | (3, 1) | (12,4) | (3, 2) | (13,5) |
(3, 0) | (11,3) | (2, 3) | (8,5) | (4, 0) | (21,4) | |||
\(\mathbf{(2,4)}\) | (9,6) | |||||||
\(\mathbb {Z}_{3^3}\) | (1, 1, 0) | (6,3) | (2, 0, 0) | (13,2) | (1, 2, 0) | (12,4) | (1, 2, 1) | (13,5) |
(1, 1, 1) | (7,4) | (2, 0, 1) | (14,3) | (2, 0, 2) | (15,4) | |||
(1, 1, 2) | (8,5) | (2, 1, 0) | (25,3) | |||||
\(\mathbf{(1,1,3)}\) | (9,6) | |||||||
\(\mathbb {Z}_{3^4}\) | (1, 0, 1, 0) | (7,4) | (1, 1, 0, 0) | (14,3) | (1, 0, 2, 0) | (13,5) | ||
(1, 0, 1, 1) | (8,5) | (1, 1, 0, 1) | (15,4) | |||||
(2, 0, 0, 0) | (34,2) | |||||||
\(\mathbf{(1,0,1,2)}\) | (9,6) | |||||||
\(\mathbb {Z}_{3^5}\) | (1, 0, 0, 1, 0) | (8,5) | (1, 0, 1, 0, 0) | (15,4) | ||||
\(\mathbf{(1,0,0,1,1)}\) | (9,6) | |||||||
\(\mathbb {Z}_{3^6}\) | \(\mathbf{(1,0,0,0,1,0)}\) | (9,6) |
\(t=8\) | \(t=9\) | \(t=10\) | ||||
---|---|---|---|---|---|---|
\((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | |
\(\mathbb {Z}_9\) | (3, 3) | (14,6) | (3, 4) | (15,7) | (3, 5) | (16,8) |
(4, 1) | (22,5) | (4, 2) | (23,6) | (4, 3) | (24,7) | |
(2, 5) | (10,7) | (5, 0) | (36,5) | (5, 1) | (37,6) | |
(2, 6) | (11,8) | (2, 7) | (12,9) | |||
\(\mathbb {Z}_{3^3}\) | (1, 2, 2) | (14,6) | (1, 2, 3) | (15,7) | (1, 2, 4) | (16,8) |
(1, 3, 0) | (22,5) | (1, 3, 1) | (23,6) | (1, 3, 2) | (24,7) | |
(2, 0, 3) | (16,5) | (2, 0, 4) | (17,6) | (1, 4, 0) | (37,6) | |
(2, 1, 1) | (26,4) | (2, 1, 2) | (27,5) | (2, 0, 5) | (18,7) | |
(3, 0, 0) | (48,3) | (2, 2, 0) | (43,4) | (2, 1, 3) | (28,6) | |
(1, 1, 4) | (10,7) | (3, 0, 1) | (49,4) | (2, 2, 1) | (44,5) | |
(1, 1, 5) | (11,8) | (3, 0, 2) | (50,5) | |||
(3, 1, 0) | (82,4) | |||||
(1, 1, 6) | (12,9) | |||||
\(\mathbb {Z}_{3^4}\) | (1, 0, 2, 1) | (14,6) | (1, 0, 2, 2) | (15,7) | (1, 0, 2, 3) | (16,8) |
(1, 1, 0, 2) | (16,5) | (1, 0, 3, 0) | (23,6) | (1, 0, 3, 1) | (24,7) | |
(1, 1, 1, 0) | (26,4) | (1, 2, 0, 0) | (49,4) | (1, 1, 0, 4) | (18,7) | |
(2, 0, 0, 1) | (35,3) | (1, 1, 0, 3) | (17,6) | (1, 1, 1, 2) | (28,6) | |
(1, 0, 1, 3) | (10,7) | (1, 1, 1, 1) | (27,5) | (1, 1, 2, 0) | (44,5) | |
(2, 0, 0, 2) | (36,4) | (1, 2, 0, 1) | (50,5) | |||
(2, 0, 1, 0) | (64,3) | (2, 0, 0, 3) | (37,5) | |||
(1, 0, 1, 4) | (11,8) | (2, 0, 1, 1) | (65,4) | |||
(2, 1, 0, 0) | (121,3) | |||||
(1, 0, 1, 5) | (12,9) | |||||
\(\mathbb {Z}_{3^5}\) | (1, 0, 0, 2, 0) | (14,6) | (1, 0, 0, 2, 1) | (15,7) | (1, 0, 0, 2, 2) | (16,8) |
(1, 0, 1, 0, 1) | (16,5) | (1, 0, 1, 0, 2) | (17,6) | (1, 0, 0, 3, 0) | (24,7) | |
(1, 1, 0, 0, 0) | (35,3) | (1, 0, 1, 1, 0) | (27,5) | (1, 0, 1, 0, 3) | (18,7) | |
(1, 0, 0, 1, 2) | (10,7) | (1, 1, 0, 0, 1) | (36,4) | (1, 0, 1, 1, 1) | (28,6) | |
(2, 0, 0, 0, 0) | (96,2) | (1, 0, 2, 0, 0) | (50,5) | |||
(1, 0, 0, 1, 3) | (11,8) | (1, 1, 0, 0, 2) | (37,5) | |||
(1, 1, 0, 1, 0) | (65,4) | |||||
(2, 0, 0, 0, 1) | (97,3) | |||||
(1, 0, 0, 1, 4) | (12,9) | |||||
\(\mathbb {Z}_{3^6}\) | (1, 0, 0, 1, 0, 0) | (16,5) | (1, 0, 0, 0, 2, 0) | (15,7) | (1, 0, 0, 0, 2, 1) | (16,8) |
(1, 0, 0, 0, 1, 1) | (10,7) | (1, 0, 0, 1, 0, 1) | (17,6) | (1, 0, 0, 1, 0, 2) | (18,7) | |
(1, 0, 1, 0, 0, 0) | (36,4) | (1, 0, 0, 1, 1, 0) | (28,6) | |||
(1, 0, 0, 0, 1, 2) | (11,8) | (1, 0, 1, 0, 0, 1) | (37,5) | |||
(1, 1, 0, 0, 0, 0) | (97,3) | |||||
(1, 0, 0, 0, 1, 3) | (12,9) | |||||
\(\mathbb {Z}_{3^7}\) | (1, 0, 0, 0, 0, 1, 0) | (10,7) | (1, 0, 0, 0, 1, 0, 0) | (17,6) | (1, 0, 0, 0, 0, 2, 0) | (16,8) |
(1, 0, 0, 0, 0, 1, 1) | (11,8) | (1, 0, 0, 0, 1, 0, 1) | (18,7) | |||
(1, 0, 0, 1, 0, 0, 0) | (37,5) | |||||
(1, 0, 0, 0, 0, 1, 2) | (12,9) | |||||
\(\mathbb {Z}_{3^8}\) | (1, 0, 0, 0, 0, 0, 1, 0) | (11,8) | (1, 0, 0, 0, 0, 1, 0, 0) | (18,7) | ||
(1, 0, 0, 0, 0, 0, 1, 1) | (12,9) | |||||
\(\mathbb {Z}_{3^9}\) | (1, 0, 0, 0, 0, 0, 0, 1, 0) | (12,9) |
\(t=4\) | \(t=5\) | \(t=6\) | \(t=7\) | \(t=8\) | ||||||
---|---|---|---|---|---|---|---|---|---|---|
\((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | \((t_1,\ldots ,t_s)\) | (r, k) | |
\(\mathbb {Z}_{25}\) | (2, 1) | (8,3) | (2, 2) | (9,4) | (3, 1) | (23,4) | (3, 2) | (24,5) | (3, 3) | (25,6) |
(3, 0) | (22,3) | (2, 3) | (10,5) | (4, 0) | (57,4) | (4, 1) | (58,5) | |||
(2, 4) | (11,6) | (2, 5) | (12,7) | |||||||
\(\mathbb {Z}_{5^3}\) | (1, 1, 0) | (8,3) | (2, 0, 0) | (29,2) | (1, 2, 0) | (23,4) | (1, 2, 1) | (24,5) | (1, 2, 2) | (25,6) |
(1, 1, 1) | (9,4) | (2, 0, 1) | (30,3) | (2, 0, 2) | (31,4) | (1, 3, 0) | (58,5) | |||
(1, 1, 2) | (10,5) | (2, 1, 0) | (84,3) | (2, 0, 3) | (32,5) | |||||
(1, 1, 3) | (11,6) | (2, 1, 1) | (85,4) | |||||||
(3, 0, 0) | (?,3) | |||||||||
(1, 1, 4) | (12,7) | |||||||||
\(\mathbb {Z}_{5^4}\) | (1, 0, 1, 0) | (9,4) | (1, 1, 0, 0) | (30,3) | (1, 0, 2, 0) | (24,5) | (1, 0, 2, 1) | (25,6) | ||
(1, 0, 1, 1) | (10,5) | (1, 1, 0, 1) | (31,4) | (1, 1, 0, 2) | (32,5) | |||||
(2, 0, 0, 0) | (134,2) | (1, 1, 1, 0) | (85,4) | |||||||
(1, 0, 1, 2) | (11,6) | (2, 0, 0, 1) | (135,3) | |||||||
(1, 0, 1, 3) | (12,7) | |||||||||
\(\mathbb {Z}_{5^5}\) | (1, 0, 0, 1, 0) | (10,5) | (1, 0, 1, 0, 0) | (31,4) | (1, 0, 0, 2, 0) | (25,6) | ||||
(1, 0, 0, 1, 1) | (11,6) | (1, 0, 1, 0, 1) | (32,5) | |||||||
(1, 1, 0, 0, 0) | (135,3) | |||||||||
(1, 0, 0, 1, 2) | (12,7) | |||||||||
\(\mathbb {Z}_{5^6}\) | (1, 0, 0, 0, 1, 0) | (11,6) | (1, 0, 0, 1, 0, 0) | (32,5) | ||||||
(1, 0, 0, 0, 1, 1) | (12,7) | |||||||||
\(\mathbb {Z}_{5^7}\) | (1, 0, 0, 0, 0, 1, 0) | (12,7) |
3 Equivalent \(\mathbb {Z}_{p^s}\)-linear GH codes
\(t=5\) | (3,0), (2,0,0) |
\(t=7\) | (4,0), (2,1,0), (2,0,0,0) |
\(t=8\) | (3,0,0) |
\(t=9\) | (5,0), (2,2,0), (2,0,1,0), (2,0,0,0,0) |
\(t=10\) | (3,1,0), (2,1,0,0) |