Cyclic Code Definitions

Definition. A code is cyclic if it is closed under cyclic shift: \[ (c_1, c_2, \ldots, c_{n-1}, c_n) \in C \Rightarrow (c_n, c_1, c_2, \ldots, c_{n-1}) \in C \]

Definition. A cyclic linear code (i.e., a cyclic code) of length \(n\) is a subset of \(\mathbb{Z}_2[x]/(x^n+1)\) that is closed under addition and closed under multiplication by \(x\). (A cyclic code is an ideal of \(\mathbb{Z}_2[x]/(x^n+1)\).)

Cyclic Code Facts

• There is a one-to-one correspondence between the divisors of \(x^n + 1\) and the cyclic codes of length \(n\).
• If \(p(x)\) is a divisor of \(x^n + 1\) of degree \(r\), then the code words of \(\langle p(x) \rangle\) are uniquely determined by products of the form \(p(x)q(x)\), where \(q(x)\) is a polynomial of degree less than \(n-r\).
• If \(p(x)\) is a divisor of \(x^n + 1\) of degree \(r\), then \(\langle p(x) \rangle\) is a \([n, n-r]\) cyclic code.
• Let \(\langle g(x) \rangle\) be the principal ideal corresponding to a cyclic code \(C\). If \(m(x)\) is a polynomial satisfying \(m(x)g(x) = x^n+1\), then \(m(x)\) is called a parity check polynomial. If \(m(x)\) is a parity check polynomial and \(p(x)\) is a message, then \(m(x)p(x) = 0\) in \(\mathbb{Z}_2[x]/(x^n+1)\) if and only if \(p(x)\) is a codeword.

Cyclic Code Examples

Over \(\mathbb{Z}_2\), \(x^7+1 = (x+1)(x^3+x+1)(x^3+x^2+1)\).

Motivating questions

• In high school algebra, can you factor \(x^2 + 4x + 3\)?
• In high school algebra, can you factor \(x^2 + 4x + 5\)?

Review: Finite Fields

• If \(p(x)\) is an irreducible polynomial of degree \(n\) in \(\mathbb{Z}_2[x]\), then \(\mathbb{Z}_2[x]/(p(x))\) is a field with \(2^n\) elements, called \(\mbox{GF}(2^n)\).
• We think of the elements of \(\mbox{GF}(2^n)\) as polynomials, under the modulus relation \(p(x) = 0\).
• Equivalently, we can think of \(\mbox{GF}(2^n)\) as a field containing \(\mathbb{Z}_2\) and an element \(\alpha\), such that \(p(\alpha) = 0\). We say that \(\mbox{GF}(2^n)\) is an extension field of \(\mathbb{Z}_2\). Notation: \(\mbox{GF}(2^n) = \mathbb{Z}_2(\alpha)\)
  • Analogy: \(\mathbb{C}\) is an extension field of \(\mathbb{R}\), containing \(i\), a root of \(x^2 + 1\).
  • \(\mathbb{C} = \mathbb{R}(i)\)

Roots of polynomials

\(\mbox{GF}(2^n)\) is a field containing \(\mathbb{Z}_2\) and an element \(\alpha\), such that \(p(\alpha) = 0\). We say that \(\mbox{GF}(2^n)\) is an extension field of \(\mathbb{Z}_2\).

Example: \(\mbox{GF(4)} = \mathbb{Z}_2[x]/(x^2+x+1)\)

• Suppose \(\alpha\) is a root of \(x^2+x+1\).
• Then \(\alpha^2 + \alpha + 1 = 0\).
• The powers of alpha are
  • \(\alpha^1 = \alpha\)
  • \(\alpha^2 = \alpha + 1\)
  • \(\alpha^3 = \alpha^2 + \alpha = 1\)

Exploration: Powers of a root

1. Over \(\mathbb{Z}_2\), let \(\alpha\) be a root of \(x^3 + x + 1\). Compute all the powers of \(\alpha\), simplifying when possible. How many are there?

2. Over \(\mathbb{Z}_2\), let \(\alpha\) be a root of \(x^3 + x^2 + x + 1\). Compute all the powers of \(\alpha\), simplifying when possible. How many are there?

Primitive roots of unity

Facts: \(\mbox{GF}(2^n)\) contains an element \(\alpha\) such that:

• \(\alpha^{2^n-1} = 1\). (\(\alpha\) is a \((2^n-1)\)th root of unity)
• Every nonzero element of \(\mbox{GF}(2^n)\) is a power of \(\alpha\). (\(\alpha\) is primitive)
• \(\alpha\) is a root of a polynomial of degree \(n\). (a primitive polynomial)

Examples: \(\mbox{GF(4)}\), \(\mbox{GF(8)}\)

List of primitive polynomials

Factoring \(x^7 + 1\) over \(\mathbb{Z}_2\)

i1 : R = ZZ/2[x]

o1 = R

o1 : PolynomialRing

i2 : factor(x^7+1)

              3           3    2
o2 = (x + 1)(x  + x + 1)(x  + x  + 1)

o2 : Expression of class Product

Factoring \(x^7 + 1\) over \(\mbox{GF}(8)\)

i1 : R = GF(8)[x]

o1 = R

o1 : PolynomialRing

i2 : factor(x^7+1)

                         2                  2           2               2
o2 = (x + 1)(x + a)(x + a )(x + a + 1)(x + a  + a)(x + a  + a + 1)(x + a  + 1)

o2 : Expression of class Product

Which primitive polynomial is Macaulay2 using?

i1 : F = GF(8)
o1 = F
o1 : GaloisField

i2 : ambient F
        ZZ
        --[a]
         2
o2 = ----------
      3
     a  + a + 1
o2 : QuotientRing

i3 : apply(1..7, i -> a^i)
          2          2       2           2
o3 = (a, a , a + 1, a  + a, a  + a + 1, a  + 1, 1)
o3 : Sequence

Splitting \(x^n + 1\)

Fact: Let \(\alpha\) be a primitive \(n\)th root of unity in some extension field of \(\mathbb{Z}_2\). Then

\[x^n + 1 = (x+1)(x+\alpha)(x+\alpha^2)\cdots(x+\alpha^{n-1})\]

So a divisor \(g(x)\) of \(x^n+1\) (and therefore a cyclic code \(\langle g(x) \rangle\)) determines a set of powers of \(\alpha\) that are zeroes of \(g(x)\).

• Jargon: \(x^n+1\) splits over \(\mathbb{Z}_2(\alpha)\).
• Analogy: All real polynomials split over \(\mathbb{C} = \mathbb{R}(i)\).

Group Exercise

Let \(C=\langle g(x)\rangle\) be a cyclic code in \(\mathbb{Z}_2[x]/(x^{13}+1)\), and let \(\alpha\) be a primitive \(13\)th root of unity. Suppose that \(g(\alpha^4) = g(\alpha^5) = 0\).

1. Let \(p(x)\) be the parity check polynomial such that \(g(x)p(x) = x^{13} + 1\). Use the fact that \(x^{13}+1\) splits over \(\mathbb{Z}_2(\alpha)\) to explain why \(p(\alpha^4) \neq 0 \neq p(\alpha^5)\).

2. Suppose that, for some \(k\), the polynomial \(m(x) = x^k\) is a code word of weight 1. Show that this leads to a contradiction.

3. Suppose that, for some \(j,k\), the polynomial \(m(x) = x^j+ x^k\) is a code word of weight 2. Show that this leads to a contradiction.

4. Explain why the minimum distance of \(C\) is at least 3.

BCH Bound Theorem over \(\mathbb{Z}_2\)

Theorem. Let \(C = \langle g(x) \rangle\) be a cyclic code in \(\mathbb{Z}_2[x]/(x^n+1)\), for \(n\) odd. Let \(\alpha\) be a primitive \(n\)th root of unity, and suppose that for some integers \(l\) and \(\delta\), \[g(\alpha^l) = g(\alpha^{l+1}) = \cdots = g(\alpha^{l+\delta})=0.\] Then the minimum distance of \(C\) is at least \(\delta + 2\).

BCH Code of designed distance \(d\)

Let \(\alpha\) be a primitive \(n\)th root of unity.

• Factor \(x^n+1\) over \(\mathbb{Z}_2\): \(x^n+1 = f_1(x)f_2(x)\cdots f_r(x)\).
• Write each factor \(f_i(x)\) as a product of \((x + \alpha^j)\)’s.
• Choose \(g(x)\) to be a product of factors \(f_i(x)\) containing \(\delta+1\) consecutive \((x+\alpha^j)\)’s.
• By the theorem, \(\langle g(x) \rangle\) will have distance at least \(\delta + 2\).

Examples

Macaulay2 tells us that over \(\mathbb{Z}_2\),

\[x^7+1 = (x+1)(x^3+x^2+1)(x^3+x+1)\]

Furthermore, over \(\text{GF}(8)\),

\[ \begin{align} x^3+x^2+1 &= (x + \alpha^3)(x+\alpha^5)(x+\alpha^6) \\ x^3+x+1 &= (x+\alpha)(x+\alpha^2)(x+\alpha^4) \end{align} \]

Exercise. In \(\mathbb{Z}_2[x]/(x^7+1)\),

1. Find a generator for a code of designed distance 3.
2. Find a generator for a code of designed distance 4.
3. Find a generator for a code of designed distance 7.

BCH Decoding

• Error-detection of BCH codes is easy, because they are cyclic codes.
• BCH codes have good error-correcting decoding algorithms.
• To illustrate, we’ll see how to correct a single error.

Correcting 1 error of a BCH code

Let \(\langle g(x) \rangle\) be a BCH code in \(\mathbb{Z}_2[x]/(x^n+1)\) of designed distance \(d \geq 3\), so \(g(\alpha^l) = g(\alpha^{l+1})=0\) for some \(l\) and for a primitive \(n\)th root of unity \(\alpha\).

Let \(r(x) = m(x) + e(x)\), where \(m(x)\) is a code word and \(e(x)\) is an error vector.

1. Compute \(s = r(\alpha^l)\). If \(s = 0\), there is no error, or there is more than one error.
2. If \(s\neq 0\), compute \(s^{-1} r(\alpha^{l+1})\). This will be a power of \(\alpha\), and the power (plus 1) gives the location of the error.

Why it works

Let \(\langle g(x) \rangle\) be a BCH code in \(\mathbb{Z}_2[x]/(x^n+1)\) of designed distance \(d \geq 3\), so \(g(\alpha^l) = g(\alpha^{l+1})=0\) for some \(l\) and for a primitive \(n\)th root of unity \(\alpha\). Let \(r(x) = m(x) + e(x)\), where \(m(x)\) is a code word and \(e(x)\) is an error vector.

1. Compute \(s = r(\alpha^l)\). If \(s = 0\), there is no error, or there is more than one error.

\[\begin{align} s &= r(\alpha^l) \\ &= m(\alpha^l) + e(\alpha^l) \\ &= 0 + e(\alpha^l) && \mbox{ since } m(x) \in \langle g(x) \rangle \end{align}\]

Why it works

2. If \(s\neq 0\), compute \(s^{-1} r(\alpha^{l+1})\). This will be a power of \(\alpha\), and the power (plus 1) gives the location of the error.

Suppose there is one error, so \(e(x) = x^k\) for some \(k\).

\[\begin{align} s^{-1}r(\alpha^{l+1}) &= (e(\alpha^l))^{-1}e(\alpha^{l+1}) && \mbox{ by part 1} \\ &= ((\alpha^l)^k)^{-1}(\alpha^{l+1})^k \\ &= \alpha^k && \mbox{ properties of exponents} \end{align}\]

Reed-Solomon Codes and Mars

The Mars Global Surveyor (MGS) was launched in 1996 and retired in 2007. According to this journal article by a JPL scientist,

The MGS downlink to Earth incorporates a 25 W travelling wave tube amplifier and a gimballed, 1:5 m high-gain antenna with a transmit gain of 39:1 dBic. Forward error correction coding options include a \((255, 223)\) Reed-Solomon code concatenated with either a \((7, \frac{1}{2})\) or \((15, \frac{1}{6})\) convolutional code.

Codes over other fields

• So far, we have been working with codes over \(\mathbb{Z}_2\).
• We can also consider codes over other finite fields.
• Hamming distance between two code words is still the number of positions where the words differ.
• For linear codes, the weight (= number of nonzero entries) of the lightest code word is still the minimum distance of the code.

BCH Bound Theorem over \(\mathbb{Z}_2\)

Recall,

Theorem. Let \(C = \langle g(x) \rangle\) be a cyclic code in \(\mathbb{Z}_2[x]/(x^n+1)\), for \(n\) odd. Let \(\alpha\) be a primitive \(n\)th root of unity, and suppose that for some integers \(l\) and \(\delta\), \[g(\alpha^l) = g(\alpha^{l+1}) = \cdots = g(\alpha^{l+\delta})=0.\] Then the minimum distance of \(C\) is at least \(\delta + 2\).

BCH Bound Theorem over any field \(F\)

The BCH Bound Theorem generalizes to other finite fields.

Theorem. Let \(C = \langle g(x) \rangle\) be a cyclic code in \(F[x]/(x^n-1)\), where \(F\) has \(q=p^m\) elements. Assume \(p \nmid n\). Let \(\alpha\) be a primitive \(n\)th root of unity, and suppose that for some integers \(l\) and \(\delta\), \[g(\alpha^l) = g(\alpha^{l+1}) = \cdots = g(\alpha^{l+\delta})=0.\] Then the minimum distance of \(C\) is at least \(\delta + 2\).

Reed-Solomon Codes

Fact: Suppose \(F\) is finite field with \(q\) elements. If \(n = q-1\), then \(F\) contains a primitive \(n\)th root of unity \(\alpha\).

So we can construct a code in \(F[x]/(x^n-1)\) using a generating polynomial of the following form.

\[ g(x) = (x-\alpha)(x-\alpha^2)\cdots(x-\alpha^{d-1}) \]

and then the theorem tells us the minimum distance of \(\langle g(x) \rangle\) is at least \(d\).

Such \(\langle g(x) \rangle\) is called a Reed-Solomon code.

Reed-Solomon optimality

• A code in \(F[x]/(x^n-1)\) with generating polynomial \[ g(x) = (x-\alpha)(x-\alpha^2)\cdots(x-\alpha^{d-1}) \] has minimum distance at least \(d\). But \(g(x)\) has only \(d\) nonzero terms. So the minimum distance equals \(d\).
• If \(|F|=q\), then there are \(q^{n-(d-1)}\) code words, since \(g(x)\) has degree \(d-1\).
• But \(q^{n-d+1}\) is the Singleton bound. So these codes are Maximum Distance Separable (MDS).

Reed-Solomon Example: \(F = \mathbb{Z}_7\)

Consider \(F = \mathbb{Z}_7\). A Reed-Solomon code is an ideal of \(\mathbb{Z}_7[x]/(x^6-1)\).

We need a primitive 6th root of unity in \(\mathbb{Z}_7\).

outer(1:6, 1:6, function(x,y) {(x^y) %% 7})

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    1    1    1    1    1
[2,]    2    4    1    2    4    1
[3,]    3    2    6    4    5    1
[4,]    4    2    1    4    2    1
[5,]    5    4    6    2    3    1
[6,]    6    1    6    1    6    1

Reed-Solomon Example: \(F = \text{GF}(8)\)

• Recall that \(\alpha\) is a primitive 7th root of unity satisfying \(\alpha^3+\alpha+1 = 0\)
• \(\text{GF}(8) = \{0, 1, \alpha, \alpha^2, \ldots, \alpha^6\}\)
• A Reed-Solomon code is an ideal of \(\text{GF}(8)[x]/(x^7+1)\).
• A Reed-Solomon code can be generated by \(g(x) = (x+\alpha)(x+\alpha^2) \stackrel{?}{=} x^2 + \alpha^4x + \alpha^3\).

Reed-Solomon Example: \(F = \text{GF}(8)\)

Exercise. Let \(g(x) = x^2 + \alpha^4x + \alpha^3\) be the generator for a Reed-Solomon code in \(\text{GF}(8)[x]/(x^7+1)\).

1. What is the minimum distance of \(\langle g(x) \rangle\)?
2. What is the length of this code?
3. What is the dimension of this code?
4. Write down a generating matrix for this code.
5. Find a code word that’s not in your generating matrix.
6. How many code words are there in \(\langle g(x) \rangle\)?

Correcting 1 error of a Reed-Solomon code

Let \(\langle g(x) \rangle\) be a Reed-Solomon code in \(F[x]/(x^n-1)\) of distance \(d \geq 3\), so \(g(\alpha) = g(\alpha^{2})=0\) for a primitive \(n\)th root of unity \(\alpha\), where \(n+1\) is the size of \(F\).

Let \(r(x) = m(x) + e(x)\), where \(m(x)\) is a code word and \(e(x)\) is an error vector.

1. Compute \(s = r(\alpha)\). If \(s = 0\), there is no error, or there is more than one error.
2. If \(s \neq 0\), compute \(s^{-1} r(\alpha^{2})\). This will have the form \(\alpha^k\), and error will be in the \(x^k\) term.
3. The error vector is \(e(x) = s\alpha^{-k}x^k\) (why?), so the decoded word is \(r(x) - e(x)\).

\(\text{GF}(2^m)\) and bit strings

Recall that the elements of \(\text{GF}(2^m)\) can be regarded as bit strings. For example,

i1 : F = GF(8)
o1 = F
o1 : GaloisField


i3 : apply(1..7, i -> a^i)
          2          2       2           2
o3 = (a, a , a + 1, a  + a, a  + a + 1, a  + 1, 1)
o3 : Sequence

We can identify these with 010, 001, 110, 011, 111, 101, 100, and along with 000 we get all the 3-digit bit strings.

Mars Global Surveyor Code

The MGS used a \((255, 233)\) Reed-Solomon code.

• What is \(F\)?
• What is the generator?
• What is the minimum distance? How many errors can be corrected?

Good for finding corrupted Bytes

• An element of \(\text{GF}(2^8)\) is an 8-bit string, i.e., a byte.
• A Reed-Solomon code over \(\text{GF}(256)\) can correct corrupted bytes.
• If errors occur in consecutive bits (e. g., a scratch on a CD or a solar flare), this is a good thing.
• Not so good if errors are randomly distributed.