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23 Row Vectors
23.1 IsRowVector (Filter)
23.1-1 IsRowVector
23.2 Operators for Row Vectors
23.2-1 NormedRowVector
23.3 Row Vectors over Finite Fields
23.3-1 ConvertToVectorRep
23.3-2 ImmutableVector
23.3-3 NumberFFVector
23.4 Coefficient List Arithmetic
23.4-1 AddVector
23.4-2 AddCoeffs
23.4-3 MultVector
23.4-4 CoeffsMod
23.5 Shifting and Trimming Coefficient Lists
23.5-1 LeftShiftRowVector
23.5-2 RightShiftRowVector
23.5-3 ShrinkRowVector
23.5-4 RemoveOuterCoeffs
23.6 Functions for Coding Theory
23.6-1 WeightVecFFE
23.6-2 DistanceVecFFE
23.6-3 DistancesDistributionVecFFEsVecFFE
23.6-4 DistancesDistributionMatFFEVecFFE
23.6-5 AClosestVectorCombinationsMatFFEVecFFE
23.6-6 CosetLeadersMatFFE
23.7 Vectors as coefficients of polynomials
23.7-1 ValuePol
23.7-2 ProductCoeffs
23.7-3 ReduceCoeffs
23.7-4 ReduceCoeffsMod
23.7-5 PowerModCoeffs
23.7-6 ShiftedCoeffs

### 23 Row Vectors

Just as in mathematics, a vector in GAP is any object which supports appropriate addition and scalar multiplication operations (see Chapter 61). As in mathematics, an especially important class of vectors are those represented by a list of coefficients with respect to some basis. These correspond roughly to the GAP concept of row vectors.

#### 23.1 IsRowVector (Filter)

##### 23.1-1 IsRowVector

`‣ IsRowVector`( obj ) | ( category )

A row vector is a vector (see `IsVector` (31.14-14)) that is also a homogeneous list of odd additive nesting depth (see 21.12). Typical examples are lists of integers and rationals, lists of finite field elements of the same characteristic, and lists of polynomials from a common polynomial ring. Note that matrices are not regarded as row vectors, because they have even additive nesting depth.

The additive operations of the vector must thus be compatible with that for lists, implying that the list entries are the coefficients of the vector with respect to some basis.

Note that not all row vectors admit a multiplication via `*` (which is to be understood as a scalar product); for example, class functions are row vectors but the product of two class functions is defined in a different way. For the installation of a scalar product of row vectors, the entries of the vector must be ring elements; note that the default method expects the row vectors to lie in `IsRingElementList`, and this category may not be implied by `IsRingElement` (31.14-16) for all entries of the row vector (see the comment in `IsVector` (31.14-14)).

Note that methods for special types of row vectors really must be installed with the requirement `IsRowVector`, since `IsVector` (31.14-14) may lead to a rank of the method below that of the default method for row vectors (see file `lib/vecmat.gi`).

    gap> IsRowVector([1,2,3]);
    true

Because row vectors are just a special case of lists, all operations and functions for lists are applicable to row vectors as well (see Chapter 21). This especially includes accessing elements of a row vector (see 21.3), changing elements of a mutable row vector (see 21.4), and comparing row vectors (see 21.10).

Note that, unless your algorithms specifically require you to be able to change entries of your vectors, it is generally better and faster to work with immutable row vectors. See Section 12.6 for more details.

#### 23.2 Operators for Row Vectors

The rules for arithmetic operations involving row vectors are in fact special cases of those for the arithmetic of lists, as given in Section 21.11 and the following sections, here we reiterate that definition, in the language of vectors.

Note that the additive behaviour sketched below is defined only for lists in the category `IsGeneralizedRowVector` (21.12-1), and the multiplicative behaviour is defined only for lists in the category `IsMultiplicativeGeneralizedRowVector` (21.12-2).

`vec1 + vec2`

returns the sum of the two row vectors vec1 and vec2. Probably the most usual situation is that vec1 and vec2 have the same length and are defined over a common field; in this case the sum is a new row vector over the same field where each entry is the sum of the corresponding entries of the vectors.

In more general situations, the sum of two row vectors need not be a row vector, for example adding an integer vector vec1 and a vector vec2 over a finite field yields the list of pointwise sums, which will be a mixture of finite field elements and integers if vec1 is longer than vec2.

`scalar + vec`

`vec + scalar`

returns the sum of the scalar scalar and the row vector vec. Probably the most usual situation is that the elements of vec lie in a common field with scalar; in this case the sum is a new row vector over the same field where each entry is the sum of the scalar and the corresponding entry of the vector.

More general situations are for example the sum of an integer scalar and a vector over a finite field, or the sum of a finite field element and an integer vector.

    gap> [ 1, 2, 3 ] + [ 1/2, 1/3, 1/4 ];
    [ 3/2, 7/3, 13/4 ]
    gap>  [ 1/2, 3/2, 1/2 ] + 1/2;
    [ 1, 2, 1 ]

`vec1 - vec2`

`scalar - vec`

`vec - scalar`

Subtracting a vector or scalar is defined as adding its additive inverse, so the statements for the addition hold likewise.

    gap> [ 1, 2, 3 ] - [ 1/2, 1/3, 1/4 ];
    [ 1/2, 5/3, 11/4 ]
    gap> [ 1/2, 3/2, 1/2 ] - 1/2;
    [ 0, 1, 0 ]

`scalar * vec`

`vec * scalar`

returns the product of the scalar scalar and the row vector vec. Probably the most usual situation is that the elements of vec lie in a common field with scalar; in this case the product is a new row vector over the same field where each entry is the product of the scalar and the corresponding entry of the vector.

More general situations are for example the product of an integer scalar and a vector over a finite field, or the product of a finite field element and an integer vector.

    gap> [ 1/2, 3/2, 1/2 ] * 2;
    [ 1, 3, 1 ]

`vec1 * vec2`

returns the standard scalar product of vec1 and vec2, i.e., the sum of the products of the corresponding entries of the vectors. Probably the most usual situation is that vec1 and vec2 have the same length and are defined over a common field; in this case the sum is an element of this field.

More general situations are for example the inner product of an integer vector and a vector over a finite field, or the inner product of two row vectors of different lengths.

    gap> [ 1, 2, 3 ] * [ 1/2, 1/3, 1/4 ];
    23/12

For the mutability of results of arithmetic operations, see 12.6.

Further operations with vectors as operands are defined by the matrix operations, see 24.3.

##### 23.2-1 NormedRowVector

`‣ NormedRowVector`( v ) | ( attribute )

returns a scalar multiple `w = c * v` of the row vector v with the property that the first nonzero entry of w is an identity element in the sense of `IsOne` (31.10-5).

    gap> NormedRowVector( [ 5, 2, 3 ] );
    [ 1, 2/5, 3/5 ]

#### 23.3 Row Vectors over Finite Fields

GAP can use compact formats to store row vectors over fields of order at most 256, based on those used by the Meat-Axe [Rin93]. This format also permits extremely efficient vector arithmetic. On the other hand element access and assignment is significantly slower than for plain lists.

The function `ConvertToVectorRep` (23.3-1) is used to convert a list into a compressed vector, or to rewrite a compressed vector over another field. Note that this function is much faster when it is given a field (or field size) as an argument, rather than having to scan the vector and try to decide the field. Supplying the field can also avoid errors and/or loss of performance, when one vector from some collection happens to have all of its entries over a smaller field than the natural field of the problem.

##### 23.3-1 ConvertToVectorRep

`‣ ConvertToVectorRep`( list[, field] ) | ( function )

`‣ ConvertToVectorRep`( list[, fieldsize] ) | ( function )

`‣ ConvertToVectorRepNC`( list[, field] ) | ( function )

`‣ ConvertToVectorRepNC`( list[, fieldsize] ) | ( function )

Called with one argument list, `ConvertToVectorRep` converts list to an internal row vector representation if possible.

Called with a list list and a finite field field, `ConvertToVectorRep` converts list to an internal row vector representation appropriate for a row vector over field.

Instead of a field also its size fieldsize may be given.

It is forbidden to call this function unless list is a plain list or a row vector, field is a field, and all elements of list lie in field. Violation of this condition can lead to unpredictable behaviour or a system crash. (Setting the assertion level to at least 2 might catch some violations before a crash, see `SetAssertionLevel` (7.5-1).)

list may already be a compressed vector. In this case, if no field or fieldsize is given, then nothing happens. If one is given then the vector is rewritten as a compressed vector over the given field unless it has the filter `IsLockedRepresentationVector`, in which case it is not changed.

The return value is the size of the field over which the vector ends up written, if it is written in a compressed representation.

In this example, we first create a row vector and then ask GAP to rewrite it, first over `GF(2)` and then over `GF(4)`.

    gap> v := [Z(2)^0,Z(2),Z(2),0*Z(2)];
    [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ]
    gap> RepresentationsOfObject(v);
    [ "IsPlistRep", "IsInternalRep" ]
    gap> ConvertToVectorRep(v);
    2
    gap> v;
    <a GF2 vector of length 4>
    gap> ConvertToVectorRep(v,4);
    4
    gap> v;
    [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ]
    gap> RepresentationsOfObject(v);
    [ "IsDataObjectRep", "Is8BitVectorRep" ]

A vector in the special representation over `GF(2)` is always viewed as `<a GF2 vector of length ...>`. Over fields of orders 3 to 256, a vector of length 10 or less is viewed as the list of its coefficients, but a longer one is abbreviated.

Arithmetic operations (see 21.11 and the following sections) preserve the compression status of row vectors in the sense that if all arguments are compressed row vectors written over the same field and the result is a row vector then also the result is a compressed row vector written over this field.

##### 23.3-2 ImmutableVector

`‣ ImmutableVector`( field, vector[, change] ) | ( operation )

Let vector be an object for which `IsRowVector` (23.1-1) or `IsVectorObj` (26.2-1) returns `true`. In the former case, vector is a list, and `ImmutableVector` returns an immutable object for which `IsRowVector` (23.1-1) returns `true` (in particular again a list), which is equal to vector, and which is in the optimal (concerning space and runtime) representation for vectors defined over field, provided that the entries of vector lie in field. In the latter case, if vector is not in `IsRowVector` (23.1-1), `ImmutableVector` returns an immutable object that is equal to the result of `ChangedBaseDomain` (26.6-3) when this is called with vector and field.

This means that vectors obtained by several calls of `ImmutableVector` for the same field are compatible for fast arithmetic without need for field conversion.

If the input vector vector is in `IsRowVector` (23.1-1) then it might change its representation as a side effect of this function. However, one cannot rely on this side effect. Also, if vector is already immutable and the result of `ImmutableVector` has the same internal representation as vector, the result is not necessarily identical to vector.

If change is `true`, then vector may be changed to become immutable; otherwise it is copied first.

##### 23.3-3 NumberFFVector

`‣ NumberFFVector`( vec, sz ) | ( operation )

returns an integer that gives the position minus one of the finite field row vector vec in the sorted list of all row vectors over the field with sz elements in the same dimension as vec. `NumberFFVector` returns `fail` if the vector cannot be represented over the field with sz elements.

    gap> v:=[0,1,2,0]*Z(3);;
    gap> NumberFFVector(v, 3);
    21
    gap> NumberFFVector(Zero(v),3);
    0
    gap> V:=EnumeratorSorted(GF(3)^4);
    <enumerator of ( GF(3)^4 )>
    gap> V[21+1] = v;
    true

#### 23.4 Coefficient List Arithmetic

The following operations all perform arithmetic on row vectors, given as homogeneous lists of the same length, containing elements of a commutative ring.

There are two reasons for using `AddRowVector` (23.4-1) in preference to arithmetic operators. Firstly, the three argument form has no single-step equivalent. Secondly `AddRowVector` (23.4-1) changes its first argument in-place, rather than allocating a new vector to hold the result, and may thus produce less garbage.

##### 23.4-1 AddVector

`‣ AddVector`( dst, src[, mul[, from, to]] ) | ( operation )

`‣ AddRowVector`( dst, src[, mul[, from, to]] ) | ( operation )

Adds the product of src and mul to dst, changing dst. If from and to are given then only the index range `[ from .. to ]` is guaranteed to be affected. Other indices may be affected, if it is more convenient to do so. Even when from and to are given, dst and src must be row vectors of the same length.

If mul is not given either then this operation simply adds src to dst.

##### 23.4-2 AddCoeffs

`‣ AddCoeffs`( list1[, poss1], list2[, poss2[, mul]] ) | ( operation )

`AddCoeffs` adds the entries of list2`{`poss2`}`, multiplied by the scalar mul, to list1`{`poss1`}`. Unbound entries in list1 are assumed to be zero. The position of the right-most non-zero element is returned.

If the ranges poss1 and poss2 are not given, they are assumed to span the whole vectors. If the scalar mul is omitted, one is used as a default.

Note that it is the responsibility of the caller to ensure that list2 has elements at position poss2 and that the result (in list1) will be a dense list.

The function is free to remove trailing (right-most) zeros.

    gap> l:=[1,2,3,4];;m:=[5,6,7];;AddCoeffs(l,m);
    4
    gap> l;
    [ 6, 8, 10, 4 ]

##### 23.4-3 MultVector

`‣ MultVector`( list, mul ) | ( operation )

`‣ MultVectorLeft`( list, mul ) | ( operation )

`‣ MultVectorRight`( list, mul ) | ( operation )

Returns: nothing

The operation `MultVectorLeft` multiplies the entries of list by the scalar mul from the left, in-place.

The operation `MultVectorRight` multiplies the entries of list by the scalar mul from the right, in-place.

Note that `MultVector` is just a synonym for `MultVectorLeft`.

    gap> A:= FreeAssociativeAlgebraWithOne( Rationals, "a", "b", "c" );;
    gap> l:=[A.1,A.2];; MultVectorLeft(l, A.3); l;
    [ (1)*c*a, (1)*c*b ]
    gap> l:=[A.1,A.2];; MultVectorRight(l, A.3); l;
    [ (1)*a*c, (1)*b*c ]

##### 23.4-4 CoeffsMod

`‣ CoeffsMod`( list1[, len1], modulus ) | ( operation )

returns the coefficient list obtained by reducing the entries in list1 modulo modulus. After reducing it shrinks the list to remove trailing zeroes. If the optional argument len1 is used, it reduces only first len1 elements of the list.

    gap> l:=[1,2,3,4];;CoeffsMod(l,2);
    [ 1, 0, 1 ]

#### 23.5 Shifting and Trimming Coefficient Lists

The following functions change coefficient lists by shifting or trimming.

##### 23.5-1 LeftShiftRowVector

`‣ LeftShiftRowVector`( list, shift ) | ( operation )

changes list by assigning list\([i]\)`:=` list\([i+\textit{shift}]\) and removing the last shift entries of the result.

##### 23.5-2 RightShiftRowVector

`‣ RightShiftRowVector`( list, shift, fill ) | ( operation )

changes list by assigning list\([i+\textit{shift}]\)`:=` list\([i]\) and filling each of the shift first entries with fill.

##### 23.5-3 ShrinkRowVector

`‣ ShrinkRowVector`( list ) | ( operation )

removes trailing zeroes from the list list.

    gap> l:=[1,0,0];;ShrinkRowVector(l);l;
    [ 1 ]

##### 23.5-4 RemoveOuterCoeffs

`‣ RemoveOuterCoeffs`( list, coef ) | ( operation )

removes coef at the beginning and at the end of list and returns the number of elements removed at the beginning.

    gap> l:=[1,1,2,1,2,1,1,2,1];; RemoveOuterCoeffs(l,1);
    2
    gap> l;
    [ 2, 1, 2, 1, 1, 2 ]

#### 23.6 Functions for Coding Theory

The following functions perform operations on finite fields vectors considered as code words in a linear code.

##### 23.6-1 WeightVecFFE

`‣ WeightVecFFE`( vec ) | ( operation )

returns the weight of the finite field vector vec, i.e. the number of nonzero entries.

##### 23.6-2 DistanceVecFFE

`‣ DistanceVecFFE`( vec1, vec2 ) | ( operation )

returns the distance between the two vectors vec1 and vec2, which must have the same length and whose elements must lie in a common field. The distance is the number of places where vec1 and vec2 differ.

##### 23.6-3 DistancesDistributionVecFFEsVecFFE

`‣ DistancesDistributionVecFFEsVecFFE`( vecs, vec ) | ( operation )

returns the distances distribution of the vector vec to the vectors in the list vecs. All vectors must have the same length, and all elements must lie in a common field. The distances distribution is a list \(d\) of length `Length(vec)+1`, such that the value \(d[i]\) is the number of vectors in vecs that have distance \(i+1\) to vec.

##### 23.6-4 DistancesDistributionMatFFEVecFFE

`‣ DistancesDistributionMatFFEVecFFE`( mat, F, vec ) | ( operation )

returns the distances distribution of the vector vec to the vectors in the vector space generated by the rows of the matrix mat over the finite field F. The length of the rows of mat and the length of vec must be equal, and all entries must lie in F. The rows of mat must be linearly independent. The distances distribution is a list \(d\) of length `Length(vec)+1`, such that the value \(d[i]\) is the number of vectors in the vector space generated by the rows of mat that have distance \(i+1\) to vec.

##### 23.6-5 AClosestVectorCombinationsMatFFEVecFFE

`‣ AClosestVectorCombinationsMatFFEVecFFE`( mat, f, vec, cnt, stop ) | ( operation )

`‣ AClosestVectorCombinationsMatFFEVecFFECoords`( mat, f, vec, cnt, stop ) | ( operation )

These functions run through the f-linear combinations of the vectors in the rows of the matrix mat that can be written as linear combinations of exactly cnt rows (that is without using zero as a coefficient). The length of the rows of mat and the length of vec must be equal, and all elements must lie in the field f. The rows of mat must be linearly independent. `AClosestVectorCombinationsMatFFEVecFFE` returns a vector from these that is closest to the vector vec. If it finds a vector of distance at most stop, which must be a nonnegative integer, then it stops immediately and returns this vector.

`AClosestVectorCombinationsMatFFEVecFFECoords` returns a length 2 list containing the same closest vector and also a vector v with exactly cnt non-zero entries, such that v times mat is the closest vector.

##### 23.6-6 CosetLeadersMatFFE

`‣ CosetLeadersMatFFE`( mat, f ) | ( operation )

Let \(C\) be the linear code over f with check matrix mat, that is, \(C\) is the right null space of mat:

\[ C = \{ v \mid mat \cdot v = 0 \}. \]

For each syndrome \(s\), the solutions of the non-homogeneous system \(\textit{mat} * v = s\) form one coset of \(C\). Equivalently, two vectors lie in the same coset if their difference lies in \(C\), just as in group theory.

`CosetLeadersMatFFE` returns, for each coset of \(C\), a vector of smallest Hamming weight in that coset. Such a vector is called a coset leader.

If `L := CosetLeadersMatFFE( mat, f )` and `q := Size( f )`, then `L[i]` is the coset leader whose syndrome is the vector with number `i-1`; in other words, `NumberFFVector( mat * L[i], q ) = i - 1` for all valid indices `i`.

Thus the list is indexed by syndromes, not by codewords.

All rows of mat must have the same length, all entries must lie in f, and the rows of mat must be linearly independent.

#### 23.7 Vectors as coefficients of polynomials

A list of ring elements can be interpreted as a row vector or the list of coefficients of a polynomial. There are a couple of functions that implement arithmetic operations based on these interpretations. GAP contains proper support for polynomials (see 66), the operations described in this section are on a lower level.

The following operations all perform arithmetic on univariate polynomials given by their coefficient lists. These lists can have different lengths but must be dense homogeneous lists containing elements of a commutative ring. Not all input lists may be empty.

In the following descriptions we will always assume that list1 is the coefficient list of the polynomial pol1 and so forth. If length parameter leni is not given, it is set to the length of listi by default.

##### 23.7-1 ValuePol

`‣ ValuePol`( coeff, x ) | ( operation )

Let coeff be the coefficients list of a univariate polynomial \(f\), and x a ring element. Then `ValuePol` returns the value \(f(\textit{x})\).

The coefficient of \(\textit{x}^i\) is assumed to be stored at position \(i+1\) in the coefficients list.

    gap> ValuePol([1,2,3],4);
    57

##### 23.7-2 ProductCoeffs

`‣ ProductCoeffs`( list1[, len1], list2[, len2] ) | ( operation )

Let \(p1\) (and \(p2\)) be polynomials given by the first len1 (len2) entries of the coefficient list list2 (list2). If len1 and len2 are omitted, they default to the lengths of list1 and list2. This operation returns the coefficient list of the product of \(p1\) and \(p2\).

    gap> l:=[1,2,3,4];;m:=[5,6,7];;ProductCoeffs(l,m);
    [ 5, 16, 34, 52, 45, 28 ]

##### 23.7-3 ReduceCoeffs

`‣ ReduceCoeffs`( list1[, len1], list2[, len2] ) | ( operation )

Let \(p1\) (and \(p2\)) be polynomials given by the first len1 (len2) entries of the coefficient list list1 (list2). If len1 and len2 are omitted, they default to the lengths of list1 and list2. `ReduceCoeffs` changes list1 to the coefficient list of the remainder when dividing p1 by p2. This operation changes list1 which therefore must be a mutable list. The operation returns the position of the last non-zero entry of the result but is not guaranteed to remove trailing zeroes.

    gap> l:=[1,2,3,4];;m:=[5,6,7];;ReduceCoeffs(l,m);
    2
    gap> l;
    [ 64/49, -24/49, 0, 0 ]

##### 23.7-4 ReduceCoeffsMod

`‣ ReduceCoeffsMod`( list1[, len1], list2[, len2], modulus ) | ( operation )

Let \(p1\) (and \(p2\)) be polynomials given by the first len1 (len2) entries of the coefficient list list1 (list2). If len1 and len2 are omitted, they default to the lengths of list1 and list2. `ReduceCoeffsMod` changes list1 to the coefficient list of the remainder when dividing p1 by p2 modulo modulus, which must be a positive integer. This operation changes list1 which therefore must be a mutable list. The operation returns the position of the last non-zero entry of the result but is not guaranteed to remove trailing zeroes.

    gap> l:=[1,2,3,4];;m:=[5,6,7];;ReduceCoeffsMod(l,m,3);
    1
    gap> l;
    [ 1, 0, 0, 0 ]

##### 23.7-5 PowerModCoeffs

`‣ PowerModCoeffs`( list1[, len1], exp, list2[, len2] ) | ( operation )

Let \(p1\) and \(p2\) be polynomials whose coefficients are given by the first len1 resp. len2 entries of the lists list1 and list2, respectively. If len1 and len2 are omitted, they default to the lengths of list1 and list2. Let exp be a positive integer. `PowerModCoeffs` returns the coefficient list of the remainder when dividing the exp-th power of \(p1\) by \(p2\). The coefficients are reduced already while powers are computed, therefore avoiding an explosion in list length.

    gap> l:=[1,2,3,4];;m:=[5,6,7];;PowerModCoeffs(l,5,m);
    [ -839462813696/678223072849, -7807439437824/678223072849 ]

##### 23.7-6 ShiftedCoeffs

`‣ ShiftedCoeffs`( list, shift ) | ( operation )

produces a new coefficient list `new` obtained by the rule `new[i+shift]:= list[i]` and filling initial holes by the appropriate zero.

    gap> l:=[1,2,3];;ShiftedCoeffs(l,2);ShiftedCoeffs(l,-2);
    [ 0, 0, 1, 2, 3 ]
    [ 3 ]

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