I am afraid answer is trivial for large N and equal just to 2. The same argument is in How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

• one can find binom x^l-1 divisible by all polynoms v_i. That would mean answer is TWO.

Mathematical formulation Fix some vectors $v_1, v_2 \in F_2^k$ , take some number N>k and consider circulant NxN matrices $C(v_1), C(v_2) $ which are defined by these vectors (put zeros on diagonals which are further then k-th diagonal).

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by $x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight, denote this weight by $dMin(N)$.

Question How does $dMin(N)$ behave depending on N?

Comments Of course, everything depends on $v_i$, this makes the question vague, so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

I am not sure my description of C(v) is clear, let me give another. Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. Consider the operator of multiplication on this polynom acting in a factor space $F_2[x]/(x^N-1)$ - this is exactly C(v).

Numerical experiment.

I took three vectors corresponding to binary form of octal numbers

G4 = 133;       % 1+D^2+D^3+D^5+D^6

G7 = 171;       % 1+D+D^2+D^3+D^6

G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:24, here is the answer:

(E.g. 9 9 10 10 12 12 13 13 12 13 15 15 15 15 15)

I am using MatLab build-in functions so for conv.codes there is certain chance my math. interpretation does not fully correspond to them, but hope it is Okay.

Error-correcting codes formulation The question is how the minimal distance of the "tail-bited" non-recursive convolutional code behave depending on block lenght N ?

Reminder Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

Pay attention this is NOT kind of code considered in here:

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Some related question:

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

But not I fix the class of codes to be convolutional.

Fix polynoms g1(x), g2(x) over F_2[x].

Question: How to find minimum over polynoms p(x) of the:

HammingWeight(p(x) g1(x) ) + HammingWeight(p(x) g2(x) ) ?

By HammingWeight of polynom I mean number of non-zero monoms, let me denote by || *||

Trivial estimate: minimum <= || g1(x)|| + || g2(x)||. Proof just put p(x) =1.

Numerical observation: apparently there are polynoms g1,g2 where this estimate is exact. Can this be true ?

Modified question 1 If I put restriction deg(p(x)) < N with N>> deg(gi) will it change minimum ? if yes what can be said about it ?

Modified question 2 If I put restriction deg(p(x)) < N and moreover will consider multiplication in the factor F_2[x]/ (x^N-1) will it change minimum ? if yes what can be said about it ?

Error-correcting codes formulation The question is how to calculate minimal distance of non-recursive convolutional code   ?

If I put restriction deg(p) < N this corresponds to various truncations of them. In particular working with F_2[x]/ (x^N-1) corresponds to "tail-biting".

PS

This is completely rewritten version of the previous question.

I am afraid answer is trivial for large N and equal just to 2. The same argument is in How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

• one can find binom x^l-1 divisible by all polynoms v_i. That would mean answer is TWO.

Mathematical formulation Fix some vectors $v_1, v_2 \in F_2^k$ , take some number N>k and consider circulant NxN matrices $C(v_1), C(v_2) $ which are defined by these vectors (put zeros on diagonals which are further then k-th diagonal).

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by $x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight, denote this weight by $dMin(N)$.

Question How does $dMin(N)$ behave depending on N?

Comments Of course, everything depends on $v_i$, this makes the question vague, so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

I am not sure my description of C(v) is clear, let me give another. Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. Consider the operator of multiplication on this polynom acting in a factor space $F_2[x]/(x^N-1)$ - this is exactly C(v).

Numerical experiment.

I took three vectors corresponding to binary form of octal numbers

G4 = 133;       % 1+D^2+D^3+D^5+D^6

G7 = 171;       % 1+D+D^2+D^3+D^6

G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:20, here is the answer:

(E.g. 9 9 10 10 12 12 13 13 12 13 15)

Error-correcting codes formulation The question is how the minimal distance of the "tail-bited" non-recursive convolutional code behave depending on block lenght N ?

Reminder Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

Pay attention this is NOT kind of code considered in here:

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Some related question:

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

But not I fix the class of codes to be convolutional.

I am afraid answer is trivial for large N and equal just to 2. The same argument is in How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

• one can find binom x^l-1 divisible by all polynoms v_i. That would mean answer is TWO.

Mathematical formulation Fix some vectors $v_1, v_2 \in F_2^k$ , take some number N>k and consider circulant NxN matrices $C(v_1), C(v_2) $ which are defined by these vectors (put zeros on diagonals which are further then k-th diagonal).

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by $x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight, denote this weight by $dMin(N)$.

Question How does $dMin(N)$ behave depending on N?

Comments Of course, everything depends on $v_i$, this makes the question vague, so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

I am not sure my description of C(v) is clear, let me give another. Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. Consider the operator of multiplication on this polynom acting in a factor space $F_2[x]/(x^N-1)$ - this is exactly C(v).

Numerical experiment.

I took three vectors corresponding to binary form of octal numbers

G4 = 133;       % 1+D^2+D^3+D^5+D^6

G7 = 171;       % 1+D+D^2+D^3+D^6

G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:24, here is the answer:

(E.g. 9 9 10 10 12 12 13 13 12 13 15 15 15 15 15)

I am using MatLab build-in functions so for conv.codes there is certain chance my math. interpretation does not fully correspond to them, but hope it is Okay.

Error-correcting codes formulation The question is how the minimal distance of the "tail-bited" non-recursive convolutional code behave depending on block lenght N ?

Reminder Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

Pay attention this is NOT kind of code considered in here:

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Some related question:

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

But not I fix the class of codes to be convolutional.

Mathematical formulation Fix some vectors $v_1, v_2 \in F_2^k$ , take some number N>k and consider circulant NxN matrices $C(v_1), C(v_2) $ which are defined by these vectors (put zeros on diagonals which are further then k-th diagonal).

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by $x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight, denote this weight by $dMin(N)$.

Question How does $dMin(N)$ behave depending on N?

Comments Of course, everything depends on $v_i$, this makes the question vague, so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

I am not sure my description of C(v) is clear, let me give another. Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. Consider the operator of multiplication on this polynom acting in a factor space $F_2[x]/(x^N-1)$ - this is exactly C(v).

Numerical experiment.

I took three vectors corresponding to binary form of octal numbers

G4 = 133;       % 1+D^2+D^3+D^5+D^6

G7 = 171;       % 1+D+D^2+D^3+D^6

G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:20, here is the answer:

(E.g. 9 9 10 10 12 12 13 13 12 13 15)

Error-correcting codes formulation The question is how the minimal distance of the "tail-bited" non-recursive convolutional code behave depending on block lenght N ?

Reminder Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

Pay attention this is NOT kind of code considered in here:

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Some related question:

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

But not I fix the class of codes to be convolutional.

I am afraid answer is trivial for large N and equal just to 2. The same argument is in How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

• one can find binom x^l-1 divisible by all polynoms v_i. That would mean answer is TWO.

Mathematical formulation Fix some vectors $v_1, v_2 \in F_2^k$ , take some number N>k and consider circulant NxN matrices $C(v_1), C(v_2) $ which are defined by these vectors (put zeros on diagonals which are further then k-th diagonal).

Consider the map "encoder" $F_2^N \to F_2^N \oplus F_2^N$, given by $x \mapsto (C(v_1) x, C(v_2) x) $.

Find some vector $y\ne 0$ in image of "encoder" with minimal Hamming weight, denote this weight by $dMin(N)$.

Question How does $dMin(N)$ behave depending on N?

Comments Of course, everything depends on $v_i$, this makes the question vague, so I am interested was it known about it for different $v_i$.

I can take several vectors $v_i$ not just two.

I am not sure my description of C(v) is clear, let me give another. Think of vector $v\in F_k$ as a polynom of degree $k-1$ in a obvious way. Consider the operator of multiplication on this polynom acting in a factor space $F_2[x]/(x^N-1)$ - this is exactly C(v).

Numerical experiment.

I took three vectors corresponding to binary form of octal numbers

G4 = 133;       % 1+D^2+D^3+D^5+D^6

G7 = 171;       % 1+D+D^2+D^3+D^6

G5 = 165;       %165  % 

(this is some practically used error-correcting code).

And calculated minDistance for N=10:20, here is the answer:

(E.g. 9 9 10 10 12 12 13 13 12 13 15)

Error-correcting codes formulation The question is how the minimal distance of the "tail-bited" non-recursive convolutional code behave depending on block lenght N ?

Reminder Non-recursive convolutional codes corresponds to semi-infinite Toeplitz matrices. "Tail-biting" is cuting this semi-infinte matrix to NxN circulant matrix.

Pay attention this is NOT kind of code considered in here:

How many k-nomials of deg N divisible by X^16+x^12+x^5 +1 ? (Spectrum of CRC-16-CCITT erroc-correcting code ?)

Some related question:

What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

But not I fix the class of codes to be convolutional.