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I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases. Another interesting paper is Perkins--Sakhonivich--Smith.

Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.

update: Turbo mentioned a work of Lenstra in the comment. It already uses the term "mixed codes" on the first page.

I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases.

Another interesting paper is Perkins--Sakhonivich--Smith. It seems to be initially cited as "mixed codes: bounds, constructions and some applications" before publication, which confused me. Fujiwara also find this reference.

Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.

update: Turbo mentioned a work of Lenstra in the comment. It already uses the term "mixed codes" on the first page.

I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases. Another interesting paper is Perkins--Sakhonivich--Smith.

Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.

I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases. Another interesting paper is Perkins--Sakhonivich--Smith.

Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.

update: Turbo mentioned a work of Lenstra in the comment. It already uses the term "mixed codes" on the first page.

I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code.

I think they also talked about some general cases. Anyway, more papers can be found from the references therein or by the key word. I also find this online list with a 4/3/2 covering code and many references.

There is a mysterious paper titled "Mixed Codes: Bounds, Constructions and Some Applications" by Perkins--Sakhnovich--Smith, appearently "submitted for publication", but I can not find it.

I think what you are looking for is mixed codes.

A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases. Another interesting paper is Perkins--Sakhonivich--Smith.

Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.