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Hand-in-Hand Teaching Guide to key in Kcmo Codes Information Bulletin No 110 Part A 2012

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Kcmo Codes Information Bulletin No 110 Part A 2012 Demand Assistance

okay all right so let's begin the next.lecture the last thing we saw in the.previous class was about minimum.distance but I want to recap what we saw.in the previous lectures let's do a.quick recap ok so right now we are.thinking of three parameters for any.code nkd code okay so once again clearly.this is going to be a K dimensional.subspace of F 2 n it's no change in that.but what is this new parameter D no yes.minimum over u not equal to 0 u in the.code the Hamming weight of you okay so.you look at all the non zero vectors in.your code the one that has smallest pick.you pick the smallest Hamming weight.among all those things you get the.minimum weight okay so that's the that's.what we saw and then this minimum.distance D like I pointed out it's not a.very linear algebraic parameter it's.more of a combinatorial parameter so.it's more difficult to determine.directly the only relation it has with.with the path with the parity check.matrix is the following relationship you.can also show da star what it is the.minimum.number of columns of H H that add to.zero.okay well the zero vector okay so I.guess it's clear so if I say the N Co.nkd code C has generator matrix G and.parity check matrix H okay remember G is.a K cross n matrix full rank and HS er n.minus K cross n matrix okay you can.think of them in systematic form if she.becomes IP H will be P transpose I K and.then the relationship gh transpose.equals 0 is also set okay.so these are all the things that we saw.in the last class like I pointed out.this is hard to determine in general.okay so usually the strategy is to.design a parity check matrix H and then.show some D minus 1 when any D minus 1.or fewer columns are linearly.independent if you do that then your.minimum distances is at least deep okay.so that's the strategy that's used to.show so if you want to show it's exactly.equal to D you also have to find the set.of D columns which do definitely add up.to 0 so you do that you can find these.things ok so let's see a couple of quick.examples I'm going to throw out a couple.of parity check matrices at you and then.ask you for the minimum distance we also.started on the Hamming code I think so.so we'll come back to that soon enough.but before that let's just quickly see.two quick examples just to drive home.this point of how easy it might be to.find the minimum distance okay so yes.the first example.okay so try that the next example you.might want to try.okay so let's see the first example I.have a parity check matrix okay the.easiest thing to determine is what.what's the first easiest thing to.determine n is the easiest parameter.that you can determine okay oh for n.minus K so okay six I'm sorry see even.the easiest parameter can sometimes be.half so N equals six and then the next.thing that you can determine is from the.parity check matrix the number of.linearly independent rows and that would.be n minus K in this case n minus K is.what okay so in this case the form of.the matrix is easy enough that you can.quickly see that the three rows are.linearly independent okay so you have an.ie sitting inside it so from there you.can quickly see that n minus K is three.so from there you can quickly conclude.that K has to be three yes also okay.what about minimum distance okay so it's.it's very safe when you compute minimum.distance to be patient and eliminate one.after the other and then go to three.okay so don't don't quickly jump to.three this case it is three answer is.correct but it's good to do it step by.step the first step is first of all it's.not B equals zero you know D is not zero.that you know what about D equals one.how do you eliminate D equals one equals.one can happen only if there is a all.zero column okay there is no all zero.column so D equals one is eliminated.cannot D cannot be one just be at least.two okay now for two what should happen.to column should be identical so it'll.repeat and you can see there is no.repetition yeah so two is also not.possible so D has to be at least three.and can you find three columns that add.to zero is the question yeah yeah so.there are multiple choices possible okay.so for instance if I pick this column.this column and this column they add to.zero okay so D clearly becomes three.okay so now another question that might.be of interest is how many minimum.weight code words are there that might.be an interesting question to think.about okay so we saw again difficult.questions but you can try to answer that.for instance in this code how many can.you quickly see.okay so the one that I put down is.clearly a minimum weight code word.another one that you can have is this.one this one and this one right so.another one you can have us this guy.this guy and this guy anything else is.anything else possible yeah.so notice there is like a non-trivial.looking minimum weight code word okay so.those three things if you add you get.what zero right yes or no okay so that's.four okay.can there be anything more so in this.case it's a really simple problem.they're only eight code words right so.you can enumerate everything and find.the answer very easily so I believe.these are the only four code words which.are of minimum weight on this okay I.also find the other ones these are the.am i right correct yes.everybody's happy okay so there are four.of them okay so it can be a bit more.devious all right so it's the only thing.I want to point out there can be some.hidden code words which suddenly show up.okay all right so let's now move on to.the other case what about the other one.okay so once again you have to follow.the same strategy okay so able to start.with D equals 1 well D equals 0 you can.just throw away just based on what you.know ok so if you want nnk let's say n.this 8 then n minus K is 4 ok so these.two things are able to see quite easily.and minus K for this easily seen because.once again there is an I sitting there.if you don't have an eye you have to do.the Gaussian elimination generate the I.and then you'll see that it works ok for.D you have to do a little bit more work.than before ok let's see.d equals 1 is definitely ruled out what.about D equals 2 but also sold out what.about D equals 3 do please nobody sure.rightly sure you have to really yeah.okay you have to check even even within.this what did I do okay so even within.these guys you have to check that no.three adds to zero and then maybe two of.them will add suddenly and they'll give.you something which is wait one it can.happen so if I make one here in this.case it won't happen but if I make one.devious looking change it can suddenly.happen for instance if I just remove.this zero and make it a one what can.happen okay so you see the last three.added adds up to zero so it's a little.bit more non-trivial to find minimum.distance but in this case you can show.no three columns will add to zero okay.so I'm not saying it's easy to check.that but you have to check all.possibilities how many possibilities you.have to check if you have to do an.exhaustive search it chose three but.clearly you don't have to do all eight.choose three right what can I avoid.easily among the first four columns.there's no way anything is going to add.up to zero okay so it's no problem there.so you can completely remove remove that.tender can see cannot completely remove.it you can take you can don't have to.take everything from there so you can.reduce it if you like you can be smart.about it.but at the end of the day it is some.kind of exhaustive checking there's.nothing much you can do okay so D will.not be three what about four one two.three four not one two three okay so.there are multiple possibilities okay so.multiple possibilities to get that for.instance very interesting interesting.looking example we can come up with if.you like yeah so if you take this guy.this guy and this guy what happens that.would be a code word right just to give.you a non-trivial example so some.strange things like this can happen so.you can get the equals four okay so so.this is the procedure think about it.very hard I mean I remember the first.time this was taught to me it I thought.about a lot of I mean algebra type of.algorithms that should give me the.minimum distance you know I mean after.all I can put it an IP and there's only.an IE there is a P and.where it was well smarter people than we.have thought of thought about it anyway.the final computation as you have to do.and always will end up being an.exhaustive computation it's very.difficult to do a smart algorithm it's.np-complete and I don't know if you are.aware of a very recent work that's out.there saying P maybe not M P and P so we.nobody will have an algorithm okay so so.so it's a difficult problem okay any.questions yes yeah yeah yeah yeah of.course I mean as you go larger and.larger than a choose 3 and n choose 3.will become better than do bucket yeah.yeah this case there are so many other.ways of doing it.anything else it's okay all right.so let's move on let me come back to.this Hamming code okay so so the code.that we saw okay so I think I think the.idea was to construct a parity check.matrices with D equals 3 right that was.the idea and we're going to pick our.rows okay and okay so this is the parity.check matrix that we're going to.construct and one of the choices for n.is to pick n equals 2 to the power R.minus 1 okay and then here you put all.nonzero binary are tuples oops.binary are tuples as as columns in order.Stroh's kids this is going to go into.columns and you get a parity check.matrix and clearly D will be three for.this okay okay what will be n minus K.will be equal to R okay so both these.things need proof proof is not too.difficult okay why will n minus K be.equal to R you can find the I are inside.this matrix okay so you will have all.those case and that will give you a.linear they give you the linear.independence that you need okay so we.get n minus K equal to R okay and then D.equals 3 you can easily see okay you add.two things you'll get another R tuple.it's just also nonzero presumably then.you also have it somewhere occurring so.you get D equals 3 okay so what are the.general parameters of this code n is 2.power R minus 1 and what is K so 2 power.R minus R minus 1 and what is D okay so.these are general binary hamming codes.binary hamming codes okay and R can be.if you want you can start with one if.you like but most non-trivial.computation starts with two so let me.just put R equals 1 2 3 x2 okay one I.think doesn't make sense so let's start.with.okay our will make K minus 1 are equal.to 1 I think we'll make K equals 0 and.our ancestors make any sense so let's.start with our equals 2 3.all right so these are the binary.Hamming codes and any D equals 3 code.you might build with binary any binary D.equals 3 code you might build they'll.look roughly like this and you have to.make if you want to make it very.efficient it will look like this right.so you know how de calls three can be.satisfied there should be no repetitions.there should be no all zero call that's.it and you get B equals three seems like.a very simple recipe and it's not too.hard okay so many codes will have D.equals three all right so that's the.Hamming code we've already seen some you.have already seen this code before but I.think it's good to see it once again.okay so so so D equals four onwards.becomes a little bit more interesting.this one equals four is not too hard it.turns out you can get a D equals four.code from a D equals three code and that.is close to optimal you can't do much.much better than that okay so so we will.see that next so what we're going to see.next is some simple modifications of.codes.okay so we will see three ideas the.first idea I'll call shortening the next.idea is puncturing the third idea is.extending okay so I have multiple.purposes in discussing these.modifications of codes the first.purposes it will give you a feel for.what these generator matrices are parity.check matrices are how do you manipulate.them how the minimum distance gets.affected and all that okay so when you.discuss this all that will become a.little bit more clear and it's a good.reinforcement of those ideas and the.other purposes these things are.extremely valuable in practice okay so.maybe not extending but shortening and.puncturing are pretty much used in all.practical applications nobody will just.use a code although it will always be a.shortened version or a punctured version.so it turns out and practice to adjust.some parameters this is very very.crucial so and from both points of view.it's useful so we will see that it's.very simple ideas but nevertheless.important to see them in some detail.okay so let's start with shortening.okay so let's say we have an nkd code.okay so C is an nkd code with generator.matrix G okay I'm dropping this.generator matrix all the time maybe it's.not a very good idea but remember that.when I say generator I mean generator.matrix yeah.suppose you have C code C given like.this okay so we know how to write this.set of all C such that C equals mg M.comes from the set of all binary K.tuples.okay so it's good to think of GES in a.systematic form at least for shortening.you always think of G it's systematic.form IP okay so then the C becomes M and.MP so what you do for the shortened.version is suppose so I'm going to now.define the shortened version we will.call it C sub s and this s is a positive.integer.okay so s is basically 1 2 3 so on so C.sub s is basically so let me write it.down carefully.set of all C so let me just write it.down a little bit differently.so so let me write the other thing first.okay so the basic idea is to restrict.your messages to have only K minus s.nonzero possibilities okay the last s.bits of the message you will always set.to zero that's the idea behind shortly.okay so it basically said okay what does.desist s tell you set last s bits of m.to zero basically you shorten your.message more than shortening the code.when you shorten the code you're.shortening your message okay so you have.the first K minus s bits as any bits.that you want and then the last s bits.you force them to zero so what will.happen when I do that when I do M times.G rows of G will pretty much drop out of.concentration I only have to worry about.the first K minus s rows so when I form.the code word what will happen the.message part of the code word will have.the first K minus s bits okay but then.the last s bits of the message will.always be zero and there's no point in.transmitting that I don't have to.transmit them so I can just simply drop.those s bits in the code also the code.word also okay.so the way you write it is okay so you.take always a little bit confusing to.write it in a very optimal way so let me.just see.maybe there is a ficient way in which.ever done it here we see.okay so this seems like the best way you.can write it so you read it as M 1 M 2.so on till M K minus s ok so you take K.minus s bits and then what do you do.leave out the 0 so the remaining things.I'm going to drop out it just doesn't.mean much so which is changing the.generator matrix a little bit and then.what do you do okay so it's a little bit.more confusing so let me just write it.let me just write it and then we'll.worry about what happens okay so maybe I.make it MP such that M must what M 1 M 2.M K minus s 1 0 it's just a laborious.way of writing but I think it's good to.write it once and see how it looks like.ok it's not clear so I can define.another P Prime if you want which is.just the first K minus s rows of P and.simply write it as M P where m is simply.the only the K minus s bits so you can.do it that way also is just another way.of writing doesn't change anything.ok is that clear reasonably clear so you.basically restrict yourself to only the.K minus s bits of the message and the.remaining bits you said to 0 this is.better question having another code with.K s mean degrees via sort utility comes.from yeah I'll come to them practical.utility is valuable let's let's come to.that slowly yeah the question was about.practical way will it be useful yeah I.mean see you don't have to think of.shortening as a great idea it's.nevertheless the simple idea ok so let's.see so the interesting question now is.what are the parameters for CS okay what.is the generator matrix for CS what is.the parity check matrix for CS those are.the things that are interesting ok so.that's the thing I want to concentrate.on a little bit like I said it gives you.practice with thinking about what these.matrices are.clearly okay so let's start with the.generator matrix for C you have AI K and.then you have okay so let me write down.the ek a bit more laborious ly 1 1 1 so.on one you have zeros here and then you.have P right so P would be B here.okay so this is the generator matrix for.C what would be a generator matrix for C.s okay maybe it is G s the last s rows.are 0 is that enough ok ok the last s.rows are 0 that's fine I agree with you.so it goes from 1 to K minus s and from.there below him I just remove okay I.remove that then what else should I.remove yeah the last s columns of I also.I should remove no that's the idea this.also should go it's quite simple to see.but anyway ok the last s columns of Y.should also be dropped okay so you would.get a GS there ok.so si s would be I K minus s and then.you would have P from 1 to K minus s ok.so you don't include said ok all right.so what about parameters ok so here you.had n KD what will be the new parameters.now as usual n is the easiest n will.become n minus s what will K be K minus.s you're able to see that immediately.right you have an IE k minus s here what.about d.what about the okay you cannot go down.okay so that's the most important thing.when you shorten D will not go down so.you will get greater than or equal to D.how do you quickly show that how do you.show that D cannot go down.more than made of water since yes.because in Syria rush I showed that the.last s bits are zero if you had a.precise proof you have to write it in.terms of contradiction okay so if does.you know suppose C S has a code of.weight code word of weight say some D.prime which is strictly less than D and.what will happen suppose C s has a C.belongs to CS and sorry CS it's nonzero.C is non zero and weight of C is equal.to some D prime which is strictly less.than D then what can I do I can now.create a C prime which will be what C 1.so on till C K minus s and then what.will I do I will add zeroes and then.what will I put for the remaining guys.CK minus s plus 1 all the way to C n.minus s now what should C prime B should.C prime be anything special no should it.belong to any code that you know of.maybe this for any code that's in this.page it should belong to C ok so this.implies C prime belongs to C why I got C.by shortening a code from the code code.word from the original code right so if.I go from a code word in the shortened.version add 0 at one point I should get.a code word in the original code and now.what can happen to weight of C prime.that's going to be D prime which.contradicts the assumption on the.minimum distance of C ok so that implies.minimum distance of C.okay let me write it down carefully.minimum distance of C unless is less.than D which is a contradiction and so.our basic assumption cannot be true you.cannot find a non zero code word in CS.which has weight less than a so simple.enough proof but you have to write it.down here to be precise about so in most.cases when you shorten it will be equal.to D okay so unless you are really lucky.or you've shortened with a lot of.foresight you cannot do it get the big.jump in minimum distance sometimes it.might happen that you get a larger.minimum distance okay okay so the.question was what is the point in doing.this it looks like you're not gaining.anything it's just designing another.linear code with different parameters n.minus K n minus SK minus s okay so it.turns out the way we do our design if.you want a particular minimum distance D.you can design it only for certain.special values of N and K not all.possible n NK is possible from the.design point of view you may not be able.to design in KD codes you will see some.design strategies later on in the course.you'll see that those strategies work.only for certain specific values of N.and K right maybe that's what that's how.it works and then you might want some.some other N and K which is very close.to the N and K I mean maybe you want n.minus s K minus s so then how do you do.it you do it by shortening so that's.that's how it is used in practice so you.design a larger code and then shorten it.to gate to get what you want okay so.that's used all the time ok all right so.what about the parity check matrix this.is a more interesting question okay.suppose I have a code C which has a.parity check matrix which looks like.this okay let's say P transpose and then.I n minus K right so this is for the.original code what will happen to the.parity check matrix of the shortened.code.okay first of all let's take let's take.a small minor step first what will be.the dimensions of the parity check.matrix of the shortened code n minus K.cross n minus s so clearly what does.this suggest a thing in the rows will go.away okay so you don't have to keep.removing anything from the row only the.columns have to go so which columns will.have to go the last l x SS columns of P.transpose happen book so that's the idea.okay so you will just look at the last s.columns of P transpose and it'll go okay.remove the last s columns we get.said okay so if you're not able to.quickly see it I will suggest another.way of thinking about the parity check.matrix so far for the encoder we've.always been using the generator matrix.you can also use the parity check matrix.for encoding it's nothing stops you from.doing that.see you have h equals let's say P.transpose ie okay the columns of the.parity check matrix are associated with.the bits of the code word so you.remember what is the condition you want.H times C transpose to be 0 so if you.have CK here CK plus 1 all the way to CN.this has to be equal to 0 right so I can.associate each of these bits with the.columns of with the columns of the.parity check matrix ok if I have a valid.codeword what should happen these.columns will be multiplied by those bits.on top and I have to add them up I.should get 0 ok remember this is my.message part okay so once I fill in my.message part how will i compute my.parity part the equation directly gives.you the answer all you have to do is.simply multiply the columns by those.bits add them up whatever vector you are.end up you end up with will simply.become your parity wait that's what this.equation is right so I can write this.equation as CK plus 1 through CN as.simply being equal to what okay so write.it carefully P transpose times C 1.through CT.okay so that's exactly what this.equation means written in a different.way okay so I can load my message bits.on the on top of P transpose multiply.the columns add it all up I will get a.vector at the end n minus K length.vector that vector is exactly my parity.vector I have to just rotate it and send.it over that's what this equation means.now when shortening what am i doing.my last s message bits are zero which.means the last s columns in P transpose.simply drop or you don't have to worry.about them so this way of viewing the.parity check matrix is also useful.okay so now once again once you think in.terms of parity check matrix even the.minimum distance becomes a little bit.easier to argue okay so previously I had.to write down some contradiction proof.on all that but from the parity check.matrix it's just immediately clear okay.if you had if you had D columns of HS.adding to zero what will happen same D.columns in H will also add to see.clearly there'll be a problem so that's.another way of thinking about it's the.same argument but maybe when you stare.at it this is a easier thing to get to.the answer all right so let's move on to.the next part of it which is puncturing.okay so finished.shortening we will see puncturing okay.so puncturing is a little bit easier to.describe.okay so puncturing is basically right.idea has to drop parities okay this is a.very very powerful idea it's used a lot.in practice okay so you drop parities so.let me just describe what happens so.once again suppose C is an nkd code.again generator would be GE etc by OD.check matrix H so the punctured version.you need a once again an integer P which.would be 1 2 3 so on CP basically be.okay so I'm going to think of everything.in systematic form so G is going to be.IP and in my code the last n minus K.bits will be parity and the first K bits.will be message okay so when I drop.parities all that I do is it's going to.be C 1 C 2 so on till C n minus P such.that C 1 C 2 so on till CN minus P EC n.minus P plus 1 so on till CN such that.belongs to C.so this is the punctured version okay so.you take all the code words of C simply.drop the last P parity bits you'll.remind you you will have a will have the.same number of code words left how do.you know that no repetition will happen.yeah so I have to pick P carefully right.so I cannot go beyond n minus K I won't.go beyond n minus K so I will make sure.that the first K bits are not affected.by the functioning so definitely I won't.lose code words if I do this because the.first K bits are message bits and I'm.not puncturing any of that so they will.all be distinct in my 2 power K code.words so as long as I draw parities my.number of code words will not go down.have the same number of code words.everything a local okay so so let's.quickly see what will happen to the.generator matrix if you have AI K P what.will happen to GS g PM Sonia.so you'll have the same i-k right.nothing changes there what will happen.to P the last P columns will simply drop.puncturing is very easy to describe what.about NK D the three parameters for the.punctured version okay so this is going.to be n minus P K remains K I just.showed you why K remains K no problem.yeah so this is going to happen okay.less than or equal to D okay but I mean.it's not good to just say less than or.equal to so then it looks like I'm.suggesting it can go to zero what can I.put a greater than or equal to if you.put a greater than or equal to also you.can also say n minus P K greater than or.equal to what can I say it will be.greater than or equal to what very.simple D has to be involved in the.answer okay so what will it be greater.than or equal to simple formula can I.say D minus P do u minus P is okay no.what's wrong with you - P that's a.perfectly good enough expression.why is D minus P okay so the intuitive.argument is in the punctured in the.original code right there will be lot of.the there will be some code word with.minimum weight D D when that gets.punctured P bits are dropped the worst.case that can happen to us all P got.punctured.all P were one in the last P bits and in.that case it'll be D minus P in any.other case you are not at an advantage.so you can only cannot be worse than.this here is not equal to D means okay.so so I'll leave it as an exercise to.see what happens to the parity check.matrix in puncturing it will be a little.bit more complicated it will be like.what happened to the generator matrix.okay so you'll see when you and the.generator matrix for for shortening okay.what happens in the generator matrix for.shortening was a little bit more.complicated right nothing just get.dropped same thing will happen here in.functioning the parity check matrix.it'll be a little bit more confusing.okay but I want to quickly point out one.thing so what does shortening do to the.rate of the code okay what does.shortening and puncturing do to the rate.of the code okay.so you had an nkd code okay suppose you.do two things.you go to CS just shortened by s bits or.you go to CP which is punctured by P.bits okay so what you get here is n.minus s k minus s greater than or equal.to D what you get here is n minus P K.and then let's say greater than equal to.D minus P I'll say that positive note.okay so what happens to the rate here.okay the rate here is K by n rate here.becomes K minus s by n minus s for s.equals one to three and all what how.will this compare with K by n will be.smaller it'll be smaller than K by n you.can show that it's very easy just.cross-multiply.see some n K will cancel and see yes if.n is greater than K this will always be.less so okay so this is the condition so.when you shorten the rate goes down and.the minimum distance can possibly.increase okay then you puncture what.happens to the rate rate goes up.KK by n minus P is greater than K by n.but the minimum distance can possibly.come down okay so this is what.so puncturing is always used for.instance in any standard communication.standard that you pick up there will be.puncturing okay they will always design.a code with lower rate and puncture it.to get higher rates what could be the.advantage of doing that instead of.designing two different codes one at.some rate say rate half another at rate.one by three you'll see more standards.we'll design only one code at rate 1 by.3 and we'll puncture to get a rate half.code that's a very standard numbers is.1/2 from 1 by 3 and 1/2 are very very.standard you'll see in almost post.animals why do they do that why don't.they design 2 different codes for a rate.1/2 and rate 1 by 3 yeah so it's it's.about reducing complexity of encoders.whenever you want to put these encoders.in some devices which are say consume.battery and you want to conserve battery.you might want to have some simple.decoders as opposed to remembering a lot.of things and all that ok but.nevertheless that's not too much of a.problem because n coding is not terribly.difficult but done so decoding also.might become easier you can have just.one decoder which runs for all of these.codes so those things are advantageous.from a design point of view ok if it's.not so important then you might want to.design two different codes in fact if.you design two different codes you might.be slightly better off but it's no.you're not considerably worse off by.doing puncturing and shortcuts that's.the idea okay so the last thing they.have about 6 minutes so let me do the.last thing which is extending it's not.too difficult it's an easier thing.okay so once again you have C which is.an nkd code.okay generator G parity check matrix H.and once again I'm thinking of this.systematic form okay so these are quite.standard okay when you extend you do see.e okay II is not any integer or anything.it's just an extension denoting it as C.II okay what this has is C 1 C 2 so on.till CN and then here you have C 1 XOR C.2 XOR okay so on till writing XOR.because I want to drive home the point.it's just one bit for all c 1 c 2 CN.belonging to c this is there okay.so extending is basically the act of.adding and overall parity.that's the idea and extending okay so.the way to visualize this is you have.the list of all code words listed down.one below the other what do you do you.go to each code word compute the overall.parity and add an extra bit and put that.as the pend do that you'll get this okay.so that's the idea so the parameters.will quickly write down the parameters.original code is nkd the parameters will.be what n plus 1 K remains K because.they're adding only a parity you're not.adding any new message but okay so it's.going to be just by parity parity added.so it's just K then what DRD plus 1 when.will it be D and when will it be D plus.1 okay so the depends on whether D is.even or odd.if D is even then what will happen it.will remain by itself so there's no.point in extending a code would be even.so that seems like a bad thing to do so.one way to write it would be to say D.plus D percentage to D mod 2.okay so if D is even I get D again if D.is odd my minimum distance goes up by.one okay and the and the Lawson rate is.negligible right it's K by n plus one.and if K and n are fairly large just.don't lose anything for not losing much.you've increased the minimum distance by.one to go from odd to even okay so this.is extending a very interesting exercise.which I urge everybody to strongly try.yes get the generator and parity check.matrix for the extended code okay so.it's a little bit non-trivial it will do.some work it's not very non-trivial but.anyway it's a good thing to try okay.you'll get a good idea of what these.parity check matrices and generator.matrices look like okay right please.quickly thinking about what happens.think about.okay so that's an important exercise.they urge you to try it will get a.clearer idea of what happens okay so we.saw a few techniques now to see.shortening puncturing an extension basic.idea like I said was to get you familiar.with what the parity check matrix is.generator matrix is what it means how do.you manipulate etc hopefully that's a.little bit clearer but another point I.want to make about this nkd it's the.following which you can imagine it's.just not too difficult but still there.are two conflicting notions here one one.one is K by n which is rate okay okay.another thing is basically minimum.distance but I will normalize it just to.make sure I can compare it with with the.rate normalized minimum distance which.is B by M okay so these two will kind of.compete with each other okay I want to.increase both right in some sense like I.said minimum distance is connected to.the error correcting capability if I.have more minimum distance I can correct.more errors okay so I want to have more.D by n where do I want to have more K by.n yeah it's it's giving very obvious.answers you know then I can send more.information per second at some level so.you can get that okay so I want to get K.by n a but you can imagine there will be.some bounds you can't keep on increasing.K by n for a fixed n you can't keep on.increasing K and D eventually something.will stop - there will be some.relationship ok so there are some bounds.relating these quantities okay which is.what we will see in the in the next.lecture.okay so we will see how to relate these.two things with some very simple ideas.but your intuition will mostly be true.normal when K by n goes up D by n will.go down and that's the idea okay.so we'll stop here for now I will pick.up from here in the next slide.

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