On the Hamming and Symbol-Pair Distance of Constacyclic Codes of Length $$p^s$$ over $$\mathbb {F}_{p^m}+ u\mathbb {F}_{p^m}$$

Symbol-Triple Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over Fq+uFq+u2Fq

“…For λ = 2, many authors studied them (see, e.g., [9][10][11][12][13][14][15][16][17][18]). In particular, for a prime power length, their structure and their symbol-pair distance were completely established in [12,19].…”

Hamming Distance of Constacyclic Codes of Length ps Over F pm +uF pm +u²F pm

“…When = 2, there are a lot of literatures on constacyclic codes over rings F p m [u]/ u 2 = F p m + uF p m for various prime p and positive integers m (see, e.g., [1], [2], [4], [8], [10], [12]- [14], [17], [18], [20], [31].) In particular, structure of and Hamming distance distibution of all constacyclic codes of length p s over F p m + uF p m were completely determined in [8], [15], [22]. When = 3, in 2015, [35] determined the structure of (δ + αu 2 )-constacyclic codes of length p s over F p m [u]/ u 3 = F p m +uF p m +u 2 F p m .…”

“…No MDS constacyclic code can be obtained in this case. 4 2 7 135 No * (x − γ) 5 2 7 132 No * (x − γ) 6 2 7 129 No * (x − γ) 7 2 7 126 No * (x − γ) 8 3 7 123 No * (x − γ) 9 3 7 120 No * (x − γ) 10 3 7 117 No * (x − γ) 11 3 7 114 No * (x − γ) 12 3 7 111 No * (x − γ) 13 3 7 108 No * (x − γ) 14 3 7 105 No * (x − γ) 15 4 7 102 No * (x − γ) 16 4 7 99 No * (x − γ) 17 4 7 96 No * (x − γ) 18 4 7 93 No * (x − γ) 19 4 7 90 No * (x − γ) 20 4 7 87 No * (x − γ) 21 4 7 84 No * (x − γ) 22 5 7 81 No * (x − γ) 23 5 7 78 No * (x − γ) 24 5 7 75 No * (x − γ) 25 5 7 72 No * (x − γ) 26 5 7 69 No * (x − γ) 27 5 7 66 No * (x − γ) 28 5 7 63 No * (x − γ) 29 6 7 60 No * (x − γ) 30 6 7 57 No * (x − γ) 31 6 7 54 No * (x − γ) 32 6 7 51 No * (x − γ) 33 6 7 48 No * (x − γ) 34 6 7 45 No * (x − γ) 35 6 7 42 No * (x − γ) 36 For future work, it would be interesting to determine the symbol-pair distances of γ-constacyclic codes of length of length p s over R, and to determine MDS symbol-pair γconstacyclic codes of length p s over R.…”

Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over \({{\mathbb{F}_{p^m}[u]}/{\langle u^3\rangle}}\)

“…10] for the case of simple-root constacyclic codes. Many researchers have scrutinized symbol-pair distances over constacyclic codes since then in [5][6][7][8][9] over many years.…”

“…When a = 2, there is significant literature on constacyclic codes over rings F p m [u]/ u 2 = F p m + uF p m for various prime p and positive integers m (see, e.g., [15][16][17][18][19][20][21][22][23].). In particular, the structure of and symbol-pair distance distibution of all constacyclic codes of length p s over F p m + uF p m were completely determined in [7,8,17].…”