In coding theory, generalized minimum-distance (GMD) decoding provides an efficient algorithm for decoding concatenated codes, which is based on using an errors-and-erasures decoder for the outer code.
A naive decoding algorithm for concatenated codes can not be an optimal way of decoding because it does not take into account the information that maximum likelihood decoding (MLD) gives. In other words, in the naive algorithm, inner received codewords are treated the same regardless of the difference between their hamming distances. Intuitively, the outer decoder should place higher confidence in symbols whose inner encodings are close to the received word. David Forney in 1966 devised a better algorithm called generalized minimum distance (GMD) decoding which makes use of those information better. This method is achieved by measuring confidence of each received codeword, and erasing symbols whose confidence is below a desired value. And GMD decoding algorithm was one of the first examples of soft-decision decoders. We will present three versions of the GMD decoding algorithm. The first two will be randomized algorithms while the last one will be a deterministic algorithm.
Setup
- Hamming distance : Given two vectors
the Hamming distance between
and
, denoted by
, is defined to be the number of positions in which
and
differ. - Minimum distance: Let
be a code. The minimum distance of code
is defined to be
where ![{\displaystyle c_{1}\neq c_{2}\in C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9add2a533ca53cf46d4442fb8a76c70eb98c9639)
- Code concatenation: Given
, consider two codes which we call outer code and inner code
![{\displaystyle C_{\text{out}}=[Q]^{K}\to [Q]^{N},\qquad C_{\text{in}}:[q]^{k}\to [q]^{n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b432223854f52245cf6ae62bd998f8a6daea11f)
- and their distances are
and
. A concatenated code can be achieved by
where
Finally we will take
to be RS code, which has an errors and erasure decoder, and
, which in turn implies that MLD on the inner code will be polynomial in
time.
- Maximum likelihood decoding (MLD): MLD is a decoding method for error correcting codes, which outputs the codeword closest to the received word in Hamming distance. The MLD function denoted by
is defined as follows. For every
. - Probability density function : A probability distribution
on a sample space
is a mapping from events of
to real numbers such that
for any event
, and
for any two mutually exclusive events
and ![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
- Expected value: The expected value of a discrete random variable
is
![{\displaystyle \mathbb {E} [X]=\sum _{x}\Pr[X=x].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b00b4c96e658352175021ec4aad62193c660043)
Randomized algorithm
Consider the received word
which was corrupted by a noisy channel. The following is the algorithm description for the general case. In this algorithm, we can decode y by just declaring an erasure at every bad position and running the errors and erasure decoding algorithm for
on the resulting vector.
Randomized_Decoder
Given :
.
- For every
, compute
. - Set
. - For every
, repeat : With probability
, set
otherwise set
. - Run errors and erasure algorithm for
on
.
Theorem 1. Let y be a received word such that there exists a codeword
such that
. Then the deterministic GMD algorithm outputs
.
Note that a naive decoding algorithm for concatenated codes can correct up to
errors.
- Lemma 1. Let the assumption in Theorem 1 hold. And if
has
errors and
erasures (when compared with
) after Step 1, then ![{\displaystyle \mathbb {E} [2e'+s']<D.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/504bbe8ae8fbbd701e22fdc916b9da981a3d91c7)
Remark. If
, then the algorithm in Step 2 will output
. The lemma above says that in expectation, this is indeed the case. Note that this is not enough to prove Theorem 1, but can be crucial in developing future variations of the algorithm.
Proof of lemma 1. For every
define
This implies that
Next for every
, we define two indicator variables:
We claim that we are done if we can show that for every
:
Clearly, by definition
Further, by the linearity of expectation, we get
To prove (2) we consider two cases:
-th block is correctly decoded (Case 1),
-th block is incorrectly decoded (Case 2):
Case 1:
Note that if
then
, and
implies
and
.
Further, by definition we have
Case 2:
In this case,
and
Since
. This follows another case analysis when
or not.
Finally, this implies
In the following sections, we will finally show that the deterministic version of the algorithm above can do unique decoding of
up to half its design distance.
Modified randomized algorithm
Note that, in the previous version of the GMD algorithm in step "3", we do not really need to use "fresh" randomness for each
. Now we come up with another randomized version of the GMD algorithm that uses the same randomness for every
. This idea follows the algorithm below.
Modified_Randomized_Decoder
Given :
, pick
at random. Then every for every
:
- Set
. - Compute
. - If
, set
otherwise set
. - Run errors and erasure algorithm for
on
.
For the proof of Lemma 1, we only use the randomness to show that
In this version of the GMD algorithm, we note that
The second equality above follows from the choice of
. The proof of Lemma 1 can be also used to show
for version2 of GMD. In the next section, we will see how to get a deterministic version of the GMD algorithm by choosing
from a polynomially sized set as opposed to the current infinite set
.
Deterministic algorithm
Let
. Since for each
, we have
where
for some
. Note that for every
, the step 1 of the second version of randomized algorithm outputs the same
. Thus, we need to consider all possible value of
. This gives the deterministic algorithm below.
Deterministic_Decoder
Given :
, for every
, repeat the following.
- Compute
for
. - Set
for every
. - If
, set
otherwise set
. - Run errors-and-erasures algorithm for
on
. Let
be the codeword in
corresponding to the output of the algorithm, if any. - Among all the
output in 4, output the one closest to ![{\displaystyle \mathbf {y} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f)
Every loop of 1~4 can be run in polynomial time, the algorithm above can also be computed in polynomial time. Specifically, each call to an errors and erasures decoder of
errors takes
time. Finally, the runtime of the algorithm above is
where
is the running time of the outer errors and erasures decoder.
See also
References
- University at Buffalo Lecture Notes on Coding Theory – Atri Rudra
- MIT Lecture Notes on Essential Coding Theory – Madhu Sudan
- University of Washington – Venkatesan Guruswami
- G. David Forney. Generalized Minimum Distance decoding. IEEE Transactions on Information Theory, 12:125–131, 1966