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# Trace description and Hamming weights of irreducible constacyclic codes

• * Corresponding author: Anuradha Sharma.
• Irreducible constacyclic codes constitute an important family of error-correcting codesand have applications in space communications.In this paper, we provide a trace description of irreducible constacyclic codes of length $n$ over the finite field $\mathbb{F}_{q}$ of order $q,$ where $n$ is a positive integer and $q$ is a prime power coprime to $n.$ As an application, we determine Hamming weight distributions of some irreducible constacyclic codes of length $n$ over $\mathbb{F}_{q}.$ We also derive a weight-divisibility theorem for irreducible constacyclic codes, and obtain both lower and upper bounds on the non-zero Hamming weights in irreducible constacyclic codes. Besides illustrating our results with examples, we list some optimal irreducible constacyclic codes that attain the distance bounds given in Grassl's Table [8].

Mathematics Subject Classification: Primary: 94B15; Secondary: 11Txx.

 Citation:

• Table 1.  Weight distribution of the $q$-ary code $\mathcal{M}_1^{(m,i)}$

 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $^1\textbf{Weight distribution of the$\boldsymbol{q}$-ary code$\mathcal{M}_1^{(m,i)}$}$ 5 78 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{60}=\mathbb{A}_{65}=312$ 7 6536 6 5 3 $\mathbb{A}_{0}=1, \mathbb{A}_{5572}=\mathbb{A}_{5628}=\mathbb{A}_{5607}=39216$ 13 33988780 8 2 4 $\mathbb{A}_{0}=1, \mathbb{A}_{31376904}=\mathbb{A}_{31373810}=\mathbb{A}_{31374720}$ $=\mathbb{A}_{31371600}=203932680$ 25 6781684 6 2 3 $\mathbb{A}_{0}=1, \mathbb{A}_{6510000}=162760416, \mathbb{A}_{6511250}=81380208$ 4 21 3 2 1 $\mathbb{A}_{0}=1, \mathbb{A}_{16}=63$ 7 600 4 3 2 $\mathbb{A}_0=1, \mathbb{A}_{504}=\mathbb{A}_{525}=1200$ 3 20 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{12}=40, \mathbb{A}_{15}=40$ 3 182 6 1 2 $\mathbb{A}_0=1, \mathbb{A}_{117}=364, \mathbb{A}_{126}=364$ 3 1640 8 1 2 $\mathbb{A}_0=1, \mathbb{A}_{1080}=3280, \mathbb{A}_{1107}=3280$ 3 14762 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{9801}=29524, \mathbb{A}_{9882}=29524$ 4 29127 9 2 3 $\mathbb{A}_0=1, \mathbb{A}_{21760}=87381, \mathbb{A}_{21888}=174762$ 4 21 3 2 1 $\mathbb{A}_0=1, \mathbb{A}_{16}=63$ 5 97656 8 2 2 $\mathbb{A}_0=1, \mathbb{A}_{78000}=195312, \mathbb{A}_{78250}=195312$ 7 29412 6 3 2 $\mathbb{A}_0=1, \mathbb{A}_{25137}=58824, \mathbb{A}_{25284}=58824$ 7 1441200 8 3 2 $\mathbb{A}_0=1, \mathbb{A}_{1234800}=2882400, \mathbb{A}_{1235829}=2882400$ 7 23539604 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{20175603}=141237624, \mathbb{A}_{20178004}=141237624$ 7 12 2 3 2 $\mathbb{A}_0=1, \mathbb{A}_{9}=24, \mathbb{A}_{12}=24$ 7 57 3 0 3 $\mathbb{A}_0=1, \mathbb{A}_{45}=114, \mathbb{A}_{48}=114, \mathbb{A}_{54}=114$ 7 19 3 4 3 $\mathbb{A}_0=1, \mathbb{A}_{15}=114, \mathbb{A}_{16}=114, \mathbb{A}_{18}=114$ 7 57 3 5 1 $\mathbb{A}_0=1, \mathbb{A}_{49}=342$ 7 400 4 1 1 $\mathbb{A}_0=1, \mathbb{A}_{343}=2400$ 7 19608 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{16716}=39216, \mathbb{A}_{16821}=39216,$ $\mathbb{A}_{16884}=39216$ 7 13072 6 4 3 $\mathbb{A}_0=1, \mathbb{A}_{11144}=39216, \mathbb{A}_{11256}=39216,$ $\mathbb{A}_{11214}=39216$ 9 820 4 2 2 $\mathbb{A}_0=1, \mathbb{A}_{720}=3280, \mathbb{A}_{738}=3280$ 9 2690420 8 7 2 $\mathbb{A}_0=1, \mathbb{A}_{2391120}=21523360, \mathbb{A}_{2391849}=21523360$ 9 1345210 8 3 4 $\mathbb{A}_0=1, \mathbb{A}_{1195560}=32285040, \mathbb{A}_{1196289}=10761680$ 13 134078 6 1 3 $\mathbb{A}_0=1, \mathbb{A}_{123669}=1608936, \mathbb{A}_{123760}=1608936,$ $\mathbb{A}_{123864}=1608936$ 13 16994390 8 5 4 $\mathbb{A}_0=1, \mathbb{A}_{15688452}=203932680, \mathbb{A}_{15686905}=203932680,$ $\mathbb{A}_{15687360}=203932680, \mathbb{A}_{15685800}=203932680$ 13 1190 4 6 4 $\mathbb{A}_0=1, \mathbb{A}_{1100}=7140, \mathbb{A}_{1104}=7140, \mathbb{A}_{1110}=7140,$ $\mathbb{A}_{1080}=7140$ 16 273 3 4 1 $\mathbb{A}_0=1, \mathbb{A}_{256}=4095$ 19 60 2 15 1 $\mathbb{A}_0=1, \mathbb{A}_{57}=360$ 25 651 3 16 1 $\mathbb{A}_0=1, \mathbb{A}_{625}=15624$ 25 10172526 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{9765000}=162760416, \mathbb{A}_{9766875}=81380208$

Table 2.  Some examples of optimal irreducible constacyclic codes $\mathcal{M}_1^{(m,i)}$ over $\mathbb{F}_{q}$

 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $\boldsymbol{d}$ 3 20 4 1 2 12 4 21 3 2 1 16 4 7 3 0 3 4 5 78 4 3 2 60 5 6 2 3 1 5 7 4 2 5 2 3 7 57 3 5 1 49 7 8 2 5 1 7 7 19 3 4 3 15 9 5 2 5 2 4
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