On the small weight codewords of the functional codes C_2(Q), Q a non-singular quadric

On the small w eigh t co dew ords of the functional co des C 2 (Q), Q a non-singular quadric F. A. B. Edoukou A. Hall ez ∗ F. Ro dier L. Storme No v emb er 2, 2018 Abstract W e study the small w eight co dew ords of the f unctio nal co de C 2 (Q), with Q a non- singular quadric of PG( N , q ). W e pro v e that the small w eight co dew ords corresp ond to the i n tersections of Q with the singular quadrics of PG( N , q ) consisting of t wo h yp erplanes. W e also calculate the num b er of co dew ords ha vin g these small weig h ts. 1 In tro d uction Consider a non-singular quadric Q o f PG( N , q ). Let Q = { P 1 , . . . , P n } , where w e normalize the co ordinates o f the p oin ts P i with resp ect to the leftmost non-zero co o rdinate. Let F b e the set of all homogeneous quadratic p olynomials f ( X 0 , . . . , X N ) defined b y N + 1 v ariables. The functional co de C 2 (Q) is the linear co de C 2 (Q) = { ( f ( P 1 ) , . . . , f ( P n )) || f ∈ F ∪ { 0 }} , defined o ver F q . This linear co de has length n = | Q | and dimension k =  N + 2 2  − 1. The t hir d fundamen tal parameter of this linear code C 2 (Q) is its minim um distance d . W e determine the 5 o r 6 smallest w eigh ts of C 2 (Q) via g e ometrical argumen ts. E v- ery homogeneous quadratic p olynomial f in N + 1 v a riables defines a quadric Q ′ : f ( X 0 , . . . , X N ) = 0 . The small w eight co dew ords of C 2 (Q) correspond to the quadrics of PG( N , q ) ha ving the larg est inters ections with Q. W e pro v e tha t these small w eigh t co dew ords corresp ond to quadrics Q ′ whic h are the union of tw o hyperplanes of P G ( N , q ). Since there are differen t p ossibilities fo r the in tersection of tw o hy p erplanes with a non-singular quadric, w e determine in this w ay the 5 or 6 smallest w eights of the functional co de C 2 (Q). W e also determine the exact n umber of co dew ords ha ving the 5 or 6 smallest w eights . ∗ The research of this author is supp orted b y a resear c h gr an t of the Resea rc h council of Ghent Uni- versit y . 1 2 Quadrics i n PG ( N , q ) The non-singular quadrics in PG( N , q ) are equal to: • the non-singular parab olic quadrics Q(2 N , q ) in PG(2 N , q ) ha ving standard equation X 2 0 + X 1 X 2 + · · · + X 2 N − 1 X 2 N = 0. These quadrics contain q 2 N − 1 + · · · + q + 1 p oin ts, and the largest dimensional spaces contained in a non-singular parab olic quadric of PG(2 N , q ) ha v e dimension N − 1, • the non- s ingular hyperb olic quadrics Q + (2 N + 1 , q ) in PG(2 N + 1 , q ) ha ving standard equation X 0 X 1 + · · · + X 2 N X 2 N +1 = 0 . T hese quadrics con tain ( q N + 1)( q N +1 − 1) / ( q − 1) = q 2 N + q 2 N − 1 + · · · + q N +1 + 2 q N + q N − 1 + · · · + q + 1 p oin ts, and the largest dimensional spaces con tained in a non-singular h yp erb olic quadric of PG(2 N + 1 , q ) ha ve dimension N , • the non-singular elliptic quadrics Q − (2 N + 1 , q ) in PG(2 N + 1 , q ) ha ving standard equation f ( X 0 , X 1 ) + X 2 X 3 + · · · + X 2 N X 2 N +1 = 0, where f ( X 0 , X 1 ) is an irreducible quadratic p o ly nomial o v er F q . These quadrics con tain ( q N +1 + 1)( q N − 1) / ( q − 1) = q 2 N + q 2 N − 1 + · · · + q N +1 + q N − 1 + · · · + q + 1 p oin ts, and the largest dimensional spaces con tained in a non- s ingular elliptic quadric of PG(2 N + 1 , q ) hav e dimension N − 1. All the quadrics o f PG( N , q ), including t he non-singular quadrics, can b e described as a quadric ha ving an s -dimensional vertex π s of singular po in ts, s ≥ − 1, and ha ving a non-singular base Q N − s − 1 in an ( N − s − 1)-dimensional space sk ew to π s , denoted by π s Q N − s − 1 . W e denote the largest dimensional spaces con tained in a quadric by the gener ators of this quadric. Since we will mak e hea vily use o f the sizes of (non-)singular quadrics of PG( N , q ), we list t hese sizes explicitly . • In PG( N , q ), a quadric ha ving an ( N − 2 d − 2)-dimensional v ertex and a h yp erbo lic quadric Q + (2 d + 1 , q ) as ba se has size q N − 1 + · · · + q N − d + 2 q N − d − 1 + q N − d − 2 + · · · + q + 1 . • In PG( N , q ), a quadric having an ( N − 2 d − 2)-dimensional ve rtex and an elliptic quadric Q − (2 d + 1 , q ) as ba se has size q N − 1 + · · · + q N − d + q N − d − 2 + · · · + q + 1 . • In PG( N , q ), a quadric hav ing an ( N − 2 d − 1) - dime nsional v ertex and a parab olic quadric Q( 2 d, q ) as base has size q N − 1 + q N − 2 + · · · + q + 1 . 2 W e note that the size of a (non-)singular quadric hav ing a non- s ingular h yp erb olic quadric as base, is alw ays larger than the size of a (non-)singular quadric having a non- singular pa r abolic quadric as base, whic h is itself alw ays larger tha n the size of a (non- )singular quadric hav ing a non- s ingular elliptic quadric as base. The quadrics ha ving the larg e st size are the unio n of tw o distinct h yp erplanes of PG( N , q ), and ha v e size 2 q N − 1 + q N − 2 + · · · + q + 1. The sec ond largest quadrics in PG( N , q ) are the quadrics ha ving an ( N − 4)-dimensional v ertex a nd a non-singular 3- dimensional h yp erb olic quadric Q + (3 , q ) as base. These quadrics ha ve size q N − 1 + 2 q N − 2 + q N − 3 + · · · + q + 1. The third la rges t quadrics in PG( N , q ) ha ve an ( N − 6)-dimensional v ertex and a non-singular h yp erbo lic quadric Q + (5 , q ) as base. These quadrics hav e size q N − 1 + q N − 2 + 2 q N − 3 + q N − 4 + · · · + q + 1. As we men tio ned in the intro duc tion, the smallest we igh t co dew ords of the co de C 2 (Q) corresp ond to the largest inte rsections of Q with other quadrics Q ′ of PG( N , q ). Let V b e the in tersection of the quadric Q with the quadric Q ′ . Tw o distinct quadrics Q and Q ′ define a unique p encil of quadrics λ Q + µ Q ′ , ( λ, µ ) ∈ F 2 q \ { (0 , 0) } . Let V = Q ∩ Q ′ , then V also lies in ev ery quadric λ Q + µ Q ′ of the p encil of quadrics defined b y Q and Q ′ . A large in tersection imp lies that there is a la r ge quadric in the p encil. The q + 1 quadrics of the p encil con tain altogether | PG( N , q ) | + q | V | p oin ts, since the p oin ts of V lie in all the quadrics of the p encil and the other points of PG( N , q ) lie in exactly o ne suc h quadric. So there is a quadric in the p encil con taining at least ( | PG( N , q ) | + q | V | ) / ( q + 1 ) p oin ts. If there is a quadric in the p enc il whic h is equal to the union of t wo h yp erplanes, then w e are at the desired conclusion that the largest intersec tions of Q arise from the in tersections of Q with the quadrics whic h are the union of tw o h yp erplanes. So assume that all q + 1 quadrics in this p encil defined b y Q and Q ′ are irreducible; w e try to find a contradiction. As already mentioned ab o v e, the largest irr educible quadrics are cones with vertex PG( N − 4 , q ) and base Q + (3 , q ), and the second largest irreducible quadrics are cones with v ertex PG( N − 6 , q ) and ba s e Q + (5 , q ). Theorem 2.1 In PG( N , q ) , with N > 6 , or N = 5 and Q = Q − (5 , q ) , if | V | > q N − 2 + 3 q N − 3 + 3 q N − 4 + 2 q N − 5 + · · · + 2 q + 1 , then in the p encil of quadrics define d by Q and Q ′ , ther e is a quadric c onsi st ing of two h y p erplanes. Pro of. Supp ose that there is no quadric consisting of t w o h yp erplanes in the p encil of quadrics. If | V | > q N − 2 + 2 q N − 3 + 2 q N − 4 + q N − 5 + · · · + q + 1, then ( | PG( N , q ) | + q | V | ) / ( q + 1) > | π N − 6 Q + (5 , q ) | , so there is a singular quadric π N − 4 Q + (3 , q ) in the p encil o f quadrics. With the lines of o ne regulus of Q + (3 , q ), to gethe r with π N − 4 , w e f orm q + 1 differen t ( N − 2 )-space s π N − 2 . W e wish to hav e that at least one of these ( N − 2)-spaces inte rsects Q in t w o ( N − 3)-dimensional spaces. All p oin ts of V app ear in at least one of these ( N − 2) - dimensional spaces Π N − 2 , so for some space π N − 2 , we hav e tha t | π N − 2 ∩ V | > | V | / ( q + 1). If | V | / ( q + 1) > | π N − 6 Q + (3 , q ) | , then π N − 2 ∩ Q is the union of t wo ( N − 3)-spaces. When | V | > q N − 2 + 3 q N − 3 + 3 q N − 4 + 2 q N − 5 + · · · + 2 q + 1, then this is v alid. So π N − 2 ∩ Q = π 1 N − 3 ∪ π 2 N − 3 . 3 These tw o ( N − 3)-dimensional spaces are con ta ined in V , so b elong to Q. This means that Q m ust ha v e subspaces of dimension N − 3 . The next table sho ws that this can o nly o ccur in small dimensions. quadric dimension generator prop ert y fulfilled Q=Q + ( N = 2 N ′ + 1 , q ) N ′ N ′ ≤ 2 Q=Q − ( N = 2 N ′ + 1 , q ) N ′ − 1 N ′ ≤ 1 Q=Q( N = 2 N ′ , q ) N ′ − 1 N ′ ≤ 2 Except for the small cases for N ′ , we hav e a contradiction, so t here is a quadric consisting of tw o h yp erplanes in the p encil of quadrics defined by Q and Q ′ . ✷ Remark 2.2 First of al l we s ay something ab out the sharpness of the b ound in Th e or em 2.1. Ther efor e we r efer to [1]. In a p encil of q + 1 non-sin g u lar el liptic quadrics Q − ( N , q ) not c ontaining hyp erplanes, the size of the interse ction o f 2 quadric s is: | Q 1 ∩ Q 2 | = q N − 2 + q N − 3 + · · · + q N +1 2 + q N − 5 2 + · · · + q + 1 Since the problem is now solv ed for dimensions N up to 4 [2, 3], there is only one case still o pen. F rom no w o n, Q will b e the h yp erb olic quadric Q + (5 , q ). If | V | > q 3 + 2 q 2 + 2 q + 1 , then there is a singular quadric π N − 4 Q + (3 , q ) = L Q + (3 , q ) in the p encil of quadrics, if w e assume that there is no quadric in the p encil whic h is the union of tw o h yp e rplanes. W e f o rm solids ω 1 , . . . , ω q +1 with L and the lines of one regulus of the base Q + (3 , q ). If | V | > q 3 + 3 q 2 + 3 q + 1, | V | / ( q + 1) > | π N − 6 Q + (3 , q ) | , so t here is a solid through L of L Q + (3 , q ) interse cting Q in tw o planes. No w we hav e three different cases: 1. L ⊂ V , 2. | L ∩ V | = 1, 3. | L ∩ V | = 2. Lemma 2.3 F or Q + (5 , q ) , if | V | > q 3 + 4 q 2 + 1 and L ⊂ V , then ther e is a quadric c onsisting of two hyp erplanes in the p encil of quadrics define d by Q and Q ′ . Pro of. Ass ume that no quadric in the p encil is the union of tw o hyperplanes. Then w e ha ve already a singular quadric L Q + (3 , q ) in the p encil and there is a solid ω 1 through L in tersecting Q in 2 planes. No w L lies in one or b oth of these planes, since L ⊂ V . Ev ery p o in t of V lies in at least one of the q + 1 solids ω 1 , . . . , ω q +1 through L . Now | V | − (union o f 2 planes) > q 3 + 4 q 2 + 1 − (2 q 2 + q + 1) = q 3 + 2 q 2 − q . So one of t he q remaining solids of ω 2 , . . . , ω q +1 con tains at least | L | + q 3 + 2 q 2 − q q = q 2 + 3 q 4 p oin ts. So one solid ω 2 con tains more t ha n | Q + (3 , q ) | p oin ts of V , so ω 2 in tersects Q in the union o f t w o pla nes. One of these planes con tains L , so L lies already in tw o planes of Q + (5 , q ). No w one of the q − 1 remaining solids ω 3 , . . . , ω q +1 con tains more than q + 1 + ( q 3 + 2 q 2 − q − 2 q 2 ) / ( q − 1 ) = q 2 + 2 q + 1 p oin ts of V . Again this implies that there is a solid ω 3 in tersecting Q in the union of t w o planes, with at least one of them con ta inin g L . This giv es us at least three planes of Q + (5 , q ) through L , whic h is imp ossible. W e hav e a con tradiction. So there is a quadric consisting of 2 hy p erplanes in the p encil of quadrics defined b y Q and Q ′ . ✷ Lemma 2.4 F or Q + (5 , q ) , if | V | > q 3 + 5 q 2 + 1 , then the c ase | L ∩ V | = 1 d o es not o c cur. Pro of. Assume that no quadric in the p encil of Q and Q ′ is the union of t wo hy p erplanes . Then we hav e already a singular quadric L Q + (3 , q ) in the p encil of quadrics. In this quadric, the line L is sk ew to the solid of Q + (3 , q ). But L is a tang ent line to Q + (5 , q ) in a p oint R since L is con tained in the cone L Q + (3 , q ), but L shares o nly one p oin t with Q + (5 , q ). Using the same argumen ts as in the preceding lemma, w e pro v e that at least three solids defined b y the line L a nd lines o f one regulus of the base Q + (3 , q ) inters ect Q in t wo planes. Thes e planes all pass thro ugh R , so they lie in the ta ng e n t h yp erplane T R (Q), whic h in tersects Q in a cone with v ertex R a nd base Q + (3 , q ) ′ . Tw o suc h planes of V in the same solid of L Q + (3 , q ) thro ugh L interse ct in a line, so they define lines of the opp osite reguli of the base Q + (3 , q ) ′ of this tangen t cone. This show s that the 4- space defined by R and the base Q + (3 , q )’ shares already six planes with Q. By B ´ ezout’s theorem, the cone R Q + (3 , q )’ is contained in V . Consider a hy p erplane throug h L ; this in tersects L Q + (3 , q ) either in a cone L Q(2 , q ) or in the union of t w o solids. So the tangen t h yp erplane T R (Q) cannot interse ct L Q + (3 , q ) in a cone R Q + (3 , q ) ′ . This giv es us a contradiction. ✷ Lemma 2.5 F or Q + (5 , q ) , if | V | > q 3 + 5 q 2 − q + 1 and | L ∩ V | = 2 , then ther e is a quadric c onsis t ing of two hyp erplanes in the p en c il of quadrics define d by Q and Q ′ . Pro of. Assume that no quadric in the p e ncil defined b y Q and Q ′ is the union of t wo h yp erplanes. Then w e ha ve already a singular quadric L Q + (3 , q ) in the p encil and there is a solid ω 1 through L in tersecting Q = Q + (5 , q ) in tw o planes. Assume that L ∩ V = { R, R ′ } . L et Q + (3 , q ) L b e the p olar quadric of L w.r.t. Q + (5 , q ) and let Q + (3 , q ) L lie in t he solid π 3 . By the same coun ting arguments a s in Lemma 2.3, we know that if | V | > q 3 + 5 q 2 − 2 q + 2, then there are 3 solids h L, L i i , with i = 1 , 2 , 3, a nd all L i b elonging to the same regulus of Q + (3 , q ), inters ecting Q in 2 pla nes. F or ev ery solid h L, L i i , w e de note b y 5 ˜ L i the line tha t the 2 planes ha ve in common, and π i 1 = h R , ˜ L i i , π i 2 = h R ′ , ˜ L i i . Then ˜ L i = π i 1 ∩ π i 2 ⊂ R ⊥ ∩ R ′⊥ = π 3 , with ⊥ the p olarit y w.r.t. Q + (5 , q ). W e use the same argumen ts f or the o pp osite regulus. This gives us ag ain 3 s olids h L, M i i , i = 1 , 2 , 3, in tersecting Q in 2 planes. W e denote b y ˜ M i the line in the in tersection of these 2 planes. These lines ˜ L i and ˜ M i b elong to the h yp erbolic quadric Q + (3 , q ) L in R ⊥ ∩ R ′⊥ , whic h is the basis fo r R Q + (3 , q ) L as w ell as for R ′ Q + (3 , q ) L . The quadric R Q + (3 , q ) L shares 6 planes w ith L Q + (3 , q ). By B´ e zout’s theorem, if R Q + (3 , q ) L 6⊂ L Q + (3 , q ), then the in tersection w ould b e of degree 4, so R Q + (3 , q ) L ⊂ L Q + (3 , q ) ∩ Q . Similarly , R ′ Q + (3 , q ) L ⊂ L Q + (3 , q ) ∩ Q. The cone L Q + (3 , q ) interse cts Q in 2 tangen t cones R Q + (3 , q ) L and R ′ Q + (3 , q ) L . W e will now lo ok at the p encil of quadrics defined b y Q a nd L Q + (3 , q ) = Q ′ . Let P b e a p oin t of π 3 \ Q + (3 , q ) L . The p oints of PG(5 , q ) \ (Q ∩ Q ′ ) lie in exactly one quadric o f t he p encil defined b y Q and Q ′ . F or the p oin t P , this m ust b e the quadric con- sisting of t he t wo h yp erplanes h R, π 3 i and h R ′ , π 3 i . F o r h R, π 3 i contains a cone R Q + (3 , q ) L and the p oin t P of this quadric, so this is one p oin t to o muc h for a quadric. So one quadric o f the p encil consists of the union of 2 h yp erplanes. ✷ Corollary 2.6 F or Q + (5 , q ) , if | V | > q 3 + 5 q 2 + 1 , then the interse ction of Q + (5 , q ) with the other quadric Q ′ is e qual to the interse ction of Q + (5 , q ) with the union of two hyp erplanes. 3 Dimension 4 W e consider a pencil of quadrics λ Q + µ Q ′ in PG ( 4 , q ), with Q a non-singular parab olic quadric Q(4 , q ). Let V = Q ∩ Q ′ . If | V | > q 2 + q + 1, then there is at least one cone P Q + (3 , q ) in this p encil. Lemma 3.1 If | V | > q 2 + ( x + 1) q + 1 , then x plan es thr ough P of the sam e r e gulus of P Q + (3 , q ) interse ct Q in 2 lines. Pro of. Consider one regulus of P Q + (3 , q ). W e wish to ha v e that x planes P L , with L a line of this regulus, in tersect Q in 2 lines. So for the first plane, this means that | V | q +1 > q + 1, since ev ery p oin t of V lies in one of the q + 1 planes P L . F o r the x - th plane, w e hav e alr e ady x − 1 planes whic h interse ct Q in 2 line s. W e imp ose that | V |− ( x − 1)(2 q +1) q − x +2 > q + 1 to guarante e that the x -th plane also in tersects Q in 2 lines. This reduces to | V | > q 2 + ( x + 1) q + 1. ✷ Denote by L i the lines o f one regulus of Q + (3 , q ) and by M i the lines o f the opp osite regulus of Q + (3 , q ), with i = 1 , 2 , . . . , q + 1. Denote b y l i 1 , l i 2 , resp. m i 1 , m i 2 , the lines of Q ∩ P L i , resp. Q ∩ P M i . W e ha ve to lo ok at 2 cases now , whether P ∈ V o r whether P 6∈ V . CASE I: P ∈ V 6 Theorem 3.2 F or Q(4 , q ) , if | V | > q 2 + 6 q + 1 and P ∈ V , then V c onsists of the union of a c one P Q(2 , q ) and another 3-dimens i onal quadric. Pro of. If we consider one regulus of the ba s e P Q + (3 , q ), then, b y the preceding lemma, there are x > 5 planes each con ta ining 2 lines of V , of whic h a t least one go es t hro ugh P . This giv es us a t least x > 5 lines thro ugh P in Q(4 , q ) ∩ P Q + (3 , q ). These x lines lie on the tangen t cone P Q(2 , q ) in T P (Q(4 , q )). By B´ ezout’s theorem, since x > 5, this cone P Q ( 2 , q ) lies completely in Q(4 , q ) and in P Q + (3 , q ). Since Q(4 , q ) ∩ P Q + (3 , q ) is a n algebraic v ariet y o f degree 4 and dimension 2, and since | V | > | P Q(2 , q ) | , V is the unio n o f P Q(2 , q ) and another 3-dimensional quadric. ✷ CASE I I: P 6∈ V Theorem 3.3 F or Q(4 , q ) , if | V | > q 2 + 11 q + 1 a nd P 6∈ V , then for q > 7 , V c onsists of the union of 2 hyp erb olic quadrics. Pro of. W e use the notations intro duc ed after the pro of of Lemma 3.1. Without loss of generality , we can assume tha t the lines of P L i lying on Q in tersected b y m 11 (resp. m 12 ) a re the lines l i 1 (resp. l i 2 ), i = 1 , . . . , x . So m 11 and m 12 are b oth in tersected b y x lines of Q . The line m 21 will in tersect at least ⌈ x 2 ⌉ of the lines l i 1 . This means that m 21 has these transv ersals in common with m 11 . Assume that these lines are the lines l 11 , · · · , l ⌈ x 2 ⌉ 1 . Also m 31 has a t least ⌈ x 2 ⌉ t ransv ersals in common with m 11 . Assume t hat at least 2 of those transv ersals also in tersect m 21 , then m 11 , m 21 , m 31 define a 3- dime nsional h yp erb olic quadric Q + (3 , q ) sharing 5 lines with Q(4 , q ). Otherwise, at least x − 1 transv ersals out of the x selected transv ersals t o m 11 are in tersecting one of m 21 and m 31 . Supp ose no w that m 41 shares at least ⌈ x 2 ⌉ tr a ns v ersals with m 11 . One o f them could b e sk ew to m 21 and m 31 , but at least ⌈ x 2 ⌉ − 1 of them in tersect m 21 or m 31 . A t least x 2 − 1 2 of them in t e rsect, f o r instance, m 21 . If this is at least 2, then m 11 , m 21 , m 41 define a 3-dimensional h yp erb olic quadric Q + (3 , q ) sharing 5 lines with Q(4 , q ). Therefore, w e obtain the same conclusion that V con tains a 3- dimensional h yp erb olic quadric when x > 1 0. Lemma 3.1 implies that w e need to imp ose that | V | > q 2 + 11 q + 1. Since in b oth cases, there is a 3-dimensional hy p erbolic quadric Q + (3 , q ) sharing 5 lines with Q (4 , q ), B´ ezout’s theorem implies that Q + (3 , q ) ⊂ Q(4 , q ). So V consists of Q + (3 , q ) and another 3-dimensional quadric. The remaining lines of V are 10 sk ew lines o f planes P L i and 10 sk ew lines o f planes P M j , and these lines of V lying in P L i in tersect the lines of V lying in P M j . So these lines also for m a 3- dime nsional h yp erb olic quadric Q + (3 , q ). ✷ Theorem 3.4 F or Q(4 , q ) , if | V | > q 2 + 11 q + 1 , then ther e is a union of 2 hyp erplanes in the p encil o f quadrics d efine d b y Q and Q’. Pro of. By Theorems 3.2 and 3.3, V consists of a 3-dimensional h yp erb o lic quadric Q + (3 , q ) in a solid π 3 and another 3-dimensional quadric. Let R b e a p oin t of π 3 \ V . The p oin ts of PG(4 , q ) \ (Q ∩ Q ′ ) lie in exactly one quadric of the p encil. Let Q ′′ b e t he unique quadric in the p encil defined b y Q and Q ′ con taining R . So π 3 shares with Q ′′ a quadric 7 and an extra p oin t R , so this is one p oin t to o m uc h for a quadric, hence there is a quadric in the p encil defined b y Q and Q ′ con taining a h yp erplane , so a quadric in the p enc il defined b y tw o hy p erplanes . ✷ 4 T ables F or the standard prop erties and notatio ns on quadrics, w e refer to [4]. 4.1 The h yp erb olic quadric in PG (2 l + 1 , q ) W e kno w that the largest in tersections of a non-singular h yp e rb olic quadric Q + (2 l + 1 , q ) in PG(2 l + 1 , q ) with the other quadrics are the intersec tions of Q + (2 l + 1 , q ) with the quadrics whic h are the union of t w o h yp erplane s Π 1 and Π 2 . W e no w discuss all the differen t p ossibilitie s for the in tersections of Q + (2 l + 1 , q ) with the union of tw o h yp e rplanes. This then gives the fiv e or six smallest weigh ts of the functional codes C 2 (Q + (2 l + 1 , q )), and the num b ers of co dewords having these w eights. W e start the discussion via the (2 l − 1)-dimensional space Π 2 l − 1 = Π 1 ∩ Π 2 . The in tersection of a (2 l − 1)-dimensional space with the non-singular h yp erb olic quadric Q + (2 l + 1 , q ) in PG(2 l + 1 , q ) is either: (1) a non-singular h yp erbo lic quadric Q + (2 l − 1 , q ), (2) a cone L Q + (2 l − 3 , q ), (3) a cone P Q(2 l − 2 , q ), or (4) a non-singular elliptic quadric Q − (2 l − 1 , q ). 1. Let PG(2 l − 1 , q ) be an (2 l − 1)-dimensional space inters ecting Q + (2 l + 1 , q ) in a non- singular (2 l − 1)-dimensional h yp e rb olic quadric Q + (2 l − 1 , q ). Then PG(2 l − 1 , q ) is the p olar space of a bisecan t line to Q + (2 l + 1 , q ). Then PG(2 l − 1 , q ) lies in t wo tangen t h yp e rplanes to Q + (2 l + 1 , q ) and in q − 1 hyperplanes in t ersecting Q + (2 l + 1 , q ) in a non- s ingular para bolic quadric Q(2 l , q ). 2. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q + (2 l + 1 , q ) in a singular quadric L Q + (2 l − 3 , q ), then PG(2 l − 1 , q ) lies in t he tangent hyperplanes to Q + (2 l + 1 , q ) in the q + 1 p oin ts P of L . 3. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q + (2 l + 1 , q ) in a singular quadric P Q ( 2 l − 2 , q ), then PG(2 l − 1 , q ) lies in the tangen t hy p erplane to Q + (2 l + 1 , q ) in P , and in q hyperplanes in tersecting Q + (2 l + 1 , q ) in no n-singular parab olic quadrics Q(2 l , q ). 4. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q + (2 l + 1 , q ) in a non-singular (2 l − 1)-dimensional elliptic quadric Q − (2 l − 1 , q ), then PG(2 l − 1 , q ) lies in q + 1 hyperplanes in tersecting Q + (2 l + 1 , q ) in non- s ingular parab olic quadrics Q(2 l , q ). In the next tables, Q + (2 l − 1 , q ) and Q − (2 l − 1 , q ) denote non-singular h yp erbolic a nd elliptic quadrics in PG(2 l − 1 , q ), P Q(2 l − 2 , q ) denotes a singular quadric with ve rtex 8 the p oin t P and base a non-singular parab olic quadric in PG(2 l − 2 , q ), L Q + (2 l − 3 , q ) denotes a singular quadric with v ertex the line L and base a non-singular h yp erb olic quadric in PG(2 l − 3 , q ), Q(2 l, q ) denotes a non-singular pa rabolic quadric in PG(2 l, q ), and P Q + (2 l − 1 , q ) denotes a singular quadric with v ertex the p oin t P and base a non- singular h yp erb olic quadric in PG(2 l − 1 , q ). In T able 1, w e denote the differen t po s sibilities for t he in tersection of Q + (2 l + 1 , q ) with the union of tw o h yp erplanes . W e describ e these p ossibilities b y giving the fo r mula for calcu lating the size of the in tersection. W e mention the sizes o f the t w o quadrics whic h are the in tersection of Π 1 and Π 2 with Q + (2 l + 1 , q ), a nd w e subtract the size of the quadric whic h is the in tersection of Π 2 l − 1 = Π 1 ∩ Π 2 with Q + (2 l + 1 , q ). Π 2 l − 1 ∩ Q + (2 l + 1 , q ) | Q + (2 l + 1 , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) Q + (2 l − 1 , q ) 2 | Q(2 l , q ) | − | Q + (2 l − 1 , q ) | (1.2) Q + (2 l − 1 , q ) | P Q + (2 l − 1 , q ) | + | Q (2 l, q ) | − | Q + (2 l − 1 , q ) | (1.3) Q + (2 l − 1 , q ) 2 | P Q + (2 l − 1 , q ) | − | Q + (2 l − 1 , q ) | (2) (2.1) L Q + (2 l − 3 , q ) 2 | P Q + (2 l − 1 , q ) | − | L Q + (2 l − 3 , q ) | (3) (3.1) P Q(2 l − 2 , q ) 2 | Q(2 l , q ) | − | P Q(2 l − 2 , q ) | (3.2) P Q(2 l − 2 , q ) | Q( 2 l, q ) | + | P Q + (2 l − 1 , q ) | − | P Q(2 l − 2 , q ) | (4) (4.1) Q − (2 l − 1 , q ) 2 | Q(2 l , q ) | − | Q − (2 l − 1 , q ) | T able 1 W e no w giv e the sizes of these in tersections of Q + (2 l + 1 , q ) with the union of t w o h yp erplanes. | Q + (2 l + 1 , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l + q l − 2 + · · · + q + 1 (1.2) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + 2 q l + q l − 2 + · · · + q + 1 (1.3) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + 3 q l + q l − 2 + · · · + q + 1 (2) (2.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + 2 q l + q l − 1 + · · · + q + 1 (3) (3.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l + q l − 1 + · · · + q + 1 (3.2) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + 2 q l + q l − 1 + · · · + q + 1 (4.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l + 2 q l − 1 + q l − 2 + · · · + q + 1 T able 2 W e no w presen t in the next table the w eigh ts of the corresp onding codew ords of C 2 (Q + (2 l + 1 , q )), and the n umbers of co dew ords having these w eigh ts. 9 W eigh t Num b er of co dew ords (1.3) w 1 = q 2 l − q 2 l − 1 − q l + q l − 1 ( q 3 l + q 2 l )( q l +1 − 1) 2 (2.1)+(3.2) w 1 + q l − q l − 1 ( q 2 l +1 − q )( q l +1 − 1)( q l − 1 +1) 2( q − 1) + ( q 3 l − 1 − q l − 1 )( q l +2 − q ) (1.2) w 1 + q l ( q 3 l + q 2 l )( q l +1 − 1)( q − 1) (4.1) w 1 + 2 q l − 2 q l − 1 q 2 l +1 ( q l +1 − 1)( q l − 1)( q − 1) 4 (3.1) w 1 + 2 q l − q l − 1 ( q 3 l − 1 − q l − 1 )( q l +1 − 1)( q 2 − q ) 2 (1.1) w 1 + 2 q l ( q 3 l + q 2 l )( q l +1 − 1)( q 2 − 3 q +2) 4 T able 3 Remark 4.1 In t he c as e that q = 2 , we have that the thir d weight c oin cides with the fourth. So in that sp e cial c ase ther e ar e only five differ ent weights. 4.2 The ellipti c quadric in PG (2 l + 1 , q ) W e kno w that the largest intersec tions of a non- s ingular elliptic quadric Q − (2 l + 1 , q ) in PG(2 l + 1 , q ) with the other quadrics are the intersec tions of Q − (2 l + 1 , q ) with the quadrics whic h are the union of t w o h yp erplane s Π 1 and Π 2 . W e no w discuss all the differen t p ossibilitie s for the in tersections of Q − (2 l + 1 , q ) with the union of tw o h yp e rplanes. This then gives the fiv e or six smallest weigh ts of the functional codes C 2 (Q − (2 l + 1 , q )), and the num b ers of co dewords having these w eights. W e again start the discussion via the (2 l − 1)-dimensional space Π 2 l − 1 = Π 1 ∩ Π 2 . The in tersection o f a (2 l − 1)-dimensional space with the non-singular elliptic quadric Q − (2 l + 1 , q ) in PG(2 l + 1 , q ) is either: (1) a non-singular elliptic quadric Q − (2 l − 1 , q ), (2) a cone P Q(2 l − 2 , q ), (3) a cone L Q − (2 l − 3 , q ), or (4 ) a non- singular h yp erb olic quadric Q + (2 l − 1 , q ). 1. Let PG(2 l − 1 , q ) b e an ( 2 l − 1)-dimensional space inters ecting Q − (2 l − 1 , q ) in a non- singular (2 l − 1 )-dimens ional elliptic quadric Q − (2 l − 1 , q ). Then PG(2 l − 1 , q ) is the p olar space of a bisecan t line to Q − (2 l + 1 , q ). Then PG(2 l − 1 , q ) lies in t w o ta ng e n t h yp erplanes to Q − (2 l + 1 , q ) a nd in q − 1 h yp erplanes interse cting Q − (2 l + 1 , q ) in a non- s ingular parab olic quadric Q( 2 l, q ). 2. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q − (2 l + 1 , q ) in a singular quadric P Q(2 l − 2 , q ), then PG(2 l − 1 , q ) lies in the tang e n t hyperplane to Q − (2 l + 1 , q ) in the p oin t P , and in q h yp erplanes intersec ting Q − (2 l + 1 , q ) in non-singular parab olic quadrics Q(2 l, q ). 3. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q − (2 l + 1 , q ) in a singular quadric L Q − (2 l − 3 , q ), then PG(2 l − 1 , q ) lies in the tangen t hy p erplane to Q − (2 l + 1 , q ) in the q + 1 p oin ts P of L . 4. Let PG(2 l − 1 , q ) b e an (2 l − 1 )-dimens ional space in tersecting Q − (2 l + 1 , q ) in a non-singular (2 l − 1)-dimensional hy p erbolic quadric Q + (2 l − 1 , q ), then PG(2 l − 1 , q ) 10 lies in q + 1 hyperplanes in tersecting Q − (2 l + 1 , q ) in non- s ingular parab olic quadrics Q(2 l , q ). In the next tables, Q + (2 l − 1 , q ) and Q − (2 l − 1 , q ) denote non-singular h yp erbolic a nd elliptic quadrics in PG(2 l − 1 , q ), P Q(2 l − 2 , q ) denotes a singular quadric with v ertex the p oin t P and base a non-singular parab olic quadric in PG(2 l − 2 , q ), L Q − (2 l − 3 , q ) denotes a singular quadric with v ertex the line L a nd base a non-singular elliptic quadric in PG(2 l − 3 , q ), Q(2 l , q ) denotes a non-singular parab olic quadric in PG(2 l , q ), and P Q − (2 l − 1 , q ) denotes a singular quadric with v ertex the p oint P and base a non-singular elliptic quadric in PG(2 l − 1 , q ). In T able 4, w e denote the differen t po s sibilities for t he in tersection of Q − (2 l + 1 , q ) with the union o f tw o hy p erplanes . Π 2 l − 1 ∩ Q − (2 l + 1 , q ) | Q − (2 l + 1 , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) Q − (2 l − 1 , q ) 2 | Q(2 l , q ) | − | Q − (2 l − 1 , q ) | (1.2) Q − (2 l − 1 , q ) | P Q − (2 l − 1 , q ) | + | Q (2 l, q ) | − | Q − (2 l − 1 , q ) | (1.3) Q − (2 l − 1 , q ) 2 | P Q − (2 l − 1 , q ) | − | Q − (2 l − 1 , q ) | (2) (2.1) P Q(2 l − 2 , q ) 2 | Q(2 l , q ) | − | P Q(2 l − 2 , q ) | (2.2) P Q(2 l − 2 , q ) | Q( 2 l, q ) | + | P Q − (2 l − 1 , q ) | − | P Q(2 l − 2 , q ) | (3) (3.1) L Q − (2 l − 3 , q ) 2 | P Q − (2 l − 1 , q ) | − | L Q − (2 l − 3 , q ) | (4) (4.1) Q + (2 l − 1 , q ) 2 | Q(2 l , q ) | − | Q + (2 l − 1 , q ) | T able 4 W e no w giv e the sizes of these in tersections of Q − (2 l + 1 , q ) with the union of t w o h yp erplanes. | Q − (2 l + 1 , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l + 2 q l − 1 + q l − 2 + · · · + q + 1 (1.2) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + 2 q l − 1 + q l − 2 + · · · + q + 1 (1.3) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 − q l + 2 q l − 1 + q l − 2 + · · · + q + 1 (2) (2.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + q l + q l − 1 + · · · + q + 1 (2.2) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + q l − 1 + · · · + q + 1 (3) (3.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l +1 + q l − 1 + · · · + q + 1 (4) (4.1) 2 q 2 l − 1 + q 2 l − 2 + · · · + q l + q l − 2 + · · · + q + 1 T able 5 W e no w presen t in the next table the w eigh ts of the corresp onding codew ords of C 2 (Q − (2 l + 1 , q )), and the n umbers of co dew ords having these w eigh ts. 11 W eigh t Num b er of co dew ords (1.1) w 1 = q 2 l − q 2 l − 1 − q l − q l − 1 ( q 3 l +1 + q 2 l )( q l − 1)( q 2 − 3 q +2) 4 (2.1) w 1 + q l − 1 ( q 2 l +1 + q l )( q 2 l − 1)( q − 1) 2 (4.1) w 1 + 2 q l − 1 q 2 l +1 ( q l +1 +1)( q l +1)( q − 1) 4 (1.2) w 1 + q l ( q 3 l +1 + q 2 l )( q l − 1)( q − 1 ) (2.2)+(3.1) w 1 + q l + q l − 1 ( q 2 l + q l − 1 )( q 2 l − 1) q + ( q l +2 + q )( q 2 l − 1)( q l − 1 − 1) 2( q − 1) (1.3) w 1 + 2 q l ( q 3 l +1 + q 2 l )( q l − 1) 2 T able 6 Remark 4.2 In t he c as e that q = 2 , we have that the thir d weight c oin cides with the fourth. So in that sp e cial c ase ther e ar e only five differ ent weights. Theorem 4.3 L et X b e a non-de gene r ate quadric (hyp erb olic or el liptic) in PG(2 l + 1 , q ) wher e l ≥ 1 . A l l the weights w i of the c o de C 2 ( X ) define d on X ar e divisib l e by q l − 1 . Pro of. Let F and f be t wo forms of degree 2 in 2 l + 2 indeterminates with l ≥ 1 and N the num b er of common zeros of F and f in F 2 l +2 q . By the theorem of Ax-Ka tz [5, p. 85], N is divisible by q l − 1 since 2 l +2 − (2+2) 2 = l − 1. On the other hand, F and f are homogeneous p olynomials, therefore N − 1 is divisible b y q − 1 . Let X and Q b e the pro jectiv e quadrics associated to F and f , o ne has |X ∩ Q| = N − 1 q − 1 . Let M = N − 1 q − 1 , one has M = k q l − 1 − 1 q − 1 = k q l − 1 − 1 q − 1 + k − 1 q − 1 = k ′ q l − 1 + π l − 2 (1) where k , k ′ ∈ Z and k = k ′ ( q − 1) + 1. By the theorem of Ax-Katz [5, p. 85] again, w e get that t he num b er o f zeros of the p olynomial F in F 2 l +2 q is divisible b y q l , so that |X | = tq l − 1 q − 1 = t q l − 1 q − 1 + t − 1 q − 1 = t ′ q l + π l − 1 (2) where t , t ′ ∈ Z and t = t ′ ( q − 1) + 1. The we igh t of a co dew ord asso ciated to the quadric X is equal to: w = |X | − |X ∩ Q| = | X | − M (3) Therefore, from (1), (2 ), and ( 3 ), w e deduce that w = t ′ q l − k ′ q l − 1 + q l − 1 . ✷ 4.3 The parab olic quadric in PG (2 l , q ) W e kno w that the la rges t in tersections of a non-singular para bolic quadric Q(2 l , q ) in PG(2 l , q ) with the other quadrics are the in tersections o f Q (2 l, q ) with the quadrics whic h are the union of t wo h yp erplanes Π 1 and Π 2 . W e no w discuss all t he differen t p ossibilities for the in tersections of Q(2 l, q ) with the union of t wo h yp erplanes. This then giv es the five smallest w eights of the functional co des C 2 (Q(2 l , q )), and the n umbers of these co dew ords. 12 W e pro ceed as follows . W e start the discussion via the (2 l − 2)-dimensional space Π 2 l − 2 whic h is the in tersection of these t w o hyperplanes Π 1 and Π 2 . The in tersection of a (2 l − 2)-dimensional space with the non-singular par a bolic quadric Q(2 l, q ) in PG(2 l, q ) is either: (1) a non- singular parab olic quadric Q(2 l − 2 , q ), (2) a cone P Q + (2 l − 3 , q ), ( 3 ) a cone P Q − (2 l − 3 , q ), or (4) a cone L Q(2 l − 4 , q ). F or q o dd, we can make the discus sion via the orthogonal p olarit y corresp onding to the non-singular parab olic quadric Q(2 l , q ). F or q ev en, w e need to use another a ppro ac h, since then Q(2 l , q ) has a n ucleus N . This implies that w e ne ed to mak e a distinction b et w een t he (2 l − 2)- dime nsional spaces Π 2 l − 2 in tersecting Q(2 l , q ) in a pa r a bolic quadric Q(2 l − 2 , q ) or a quadric L Q(2 l − 4 , q ), con ta ining the nucleu s N of Q(2 l, q ), and those not con taining the n ucleus N of Q (2 l, q ). In [4], these (2 l − 2) -dimen sional spaces are resp e ctiv ely called n u cle ar and no n-nucle ar . W e first discuss the case q o dd. 1. Let PG(2 l − 2 , q ) b e an (2 l − 2)- dimensional space in tersecting Q(2 l , q ) in a non- singular (2 l − 2)-dimensional para bolic quadric Q(2 l − 2 , q ). Then PG(2 l − 2 , q ) is the p olar space of a bisecan t or external line to Q(2 l, q ). In the first case, PG(2 l − 2 , q ) lies in t wo tang ent hyperplanes to Q(2 l, q ), ( q − 1) / 2 h yp e rplanes inters ecting Q(2 l , q ) in a non- s ingular h yp erb olic quadric Q + (2 l − 1 , q ), a nd in ( q − 1) / 2 h yp erplanes in tersecting Q(2 l, q ) in a non-singular elliptic quadric Q − (2 l − 1 , q ). In the second case, PG(2 l − 2 , q ) lies in ( q + 1) / 2 hy p erplanes inters ecting Q(2 l, q ) in a non-singular h yp erb olic quadric Q + (2 l − 1 , q ), and in ( q + 1) / 2 hyperplanes inte rsecting Q(2 l, q ) in a non- s ingular elliptic quadric Q − (2 l − 1 , q ). 2. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric P Q + (2 l − 3 , q ), then PG(2 l − 2 , q ) lies in the tangen t h yp erplane to Q(2 l , q ) in P and in q h yp erplanes in tersecting Q(2 l , q ) in non-singular hyperb olic quadrics Q + (2 l − 1 , q ). 3. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric P Q − (2 l − 3 , q ), then PG(2 l − 2 , q ) lies in the tangen t h yp erplane to Q(2 l , q ) in P , and in q hy p erplanes interse cting Q(2 l , q ) in no n- sin gular elliptic quadrics Q − (2 l − 1 , q ). 4. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric L Q(2 l − 4 , q ), then PG(2 l − 2 , q ) lies in the ta ng e n t hyperplanes to Q(2 l , q ) in the q + 1 p oin ts P of L . In the next tables, Q(2 l − 2 , q ) and Q(2 l , q ) denote non- singular parab olic quadrics in PG(2 l − 2 , q ) and in PG(2 l , q ), Q + (2 l − 1 , q ) denotes a non-singular h yp erb olic quadric in PG(2 l − 1 , q ), Q − (2 l − 1 , q ) denotes a non-singular elliptic quadric in PG(2 l − 1 , q ), P Q ( 2 l − 2 , q ) denotes a singular quadric with v ertex the p oin t P and base a non-singular parab olic quadric in PG(2 l − 2 , q ), P Q + (2 l − 3 , q ) denotes a singular quadric with v ertex the p oin t P and base a non- singular h yp erb olic quadric in PG(2 l − 3 , q ), P Q − (2 l − 3 , q ) 13 denotes a singular quadric with v ertex the p oint P and base a non-singular elliptic quadric in PG(2 l − 3 , q ), and L Q(2 l − 4 , q ) denotes a singular quadric with v ertex the line L a nd base a non-singular parab olic quadric in PG(2 l − 4 , q ). In T able 7, w e denote the differen t p ossibilities for the in tersection of Q(2 l , q ) with the union of tw o h yp e rplanes. Π 2 l − 2 ∩ Q(2 l , q ) | Q(2 l , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) Q(2 l − 2 , q ) 2 | Q + (2 l − 1 , q ) | − | Q(2 l − 2 , q ) | (1.2) Q(2 l − 2 , q ) | Q + (2 l − 1 , q ) | + | Q − (2 l − 1 , q ) | − | Q(2 l − 2 , q ) | (1.3) Q(2 l − 2 , q ) | P Q(2 l − 2 , q ) | + | Q + (2 l − 1 , q ) | − | Q(2 l − 2 , q ) | (1.4) Q(2 l − 2 , q ) | P Q(2 l − 2 , q ) | + | Q − (2 l − 1 , q ) | − | Q(2 l − 2 , q ) | (1.5) Q(2 l − 2 , q ) 2 | Q − (2 l − 1 , q ) | − | Q(2 l − 2 , q ) | (1.6) Q(2 l − 2 , q ) 2 | P Q(2 l − 2 , q ) | − | Q ( 2 l − 2 , q ) | (2) (2.1) P Q + (2 l − 3 , q ) 2 | Q + (2 l − 1 , q ) | − | P Q + (2 l − 3 , q ) | (2.2) P Q + (2 l − 3 , q ) | Q + (2 l − 1 , q ) | + | P Q(2 l − 2 , q ) | − | P Q + (2 l − 3 , q ) | (3) (3.1) P Q − (2 l − 3 , q ) 2 | Q − (2 l − 1 , q ) | − | P Q − (2 l − 3 , q ) | (3.2) P Q − (2 l − 3 , q ) | Q − (2 l − 1 , q ) | + | P Q(2 l − 2 , q ) | − | P Q − (2 l − 3 , q ) | (4) (4.1) L Q(2 l − 4 , q ) 2 | P Q(2 l − 2 , q ) | − | L Q(2 l − 4 , q ) | T able 7 W e no w giv e the sizes of these intersec tions of Q(2 l, q ) with the union of tw o h yp e r- planes. | Q(2 l , q ) ∩ (Π 1 ∪ Π 2 ) | (1) (1.1) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + 3 q l − 1 + q l − 2 + · · · + q + 1 (1.2) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 1 + q l − 2 + · · · + q + 1 (1.3) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + 2 q l − 1 + q l − 2 + · · · + q + 1 (1.4) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 2 + · · · + q + 1 (1.5) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l − q l − 1 + q l − 2 + · · · + q + 1 (1.6) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 1 + q l − 2 + · · · + q + 1 (2) (2.1) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + 2 q l − 1 + q l − 2 + · · · + q + 1 (2.2) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 1 + q l − 2 + · · · + q + 1 (3) (3.1) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 2 + · · · + q + 1 (3.2) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 1 + q l − 2 + · · · + q + 1 (4) (4.1) 2 q 2 l − 2 + q 2 l − 3 + · · · + q l + q l − 1 + q l − 2 + · · · + q + 1 T able 8 14 W eigh t Num b er of co dew ords (1.1) w 1 = q 2 l − 1 − q 2 l − 2 − 2 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 )( q − 3) 16 + q 2 l − 1 ( q 2 l − 1)( q − 1) 2 16 (1.3)+(2.1) w 1 + q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 ) 2 + q l ( q l − 1 +1)( q 2 l − 1)( q − 1) 4 (1.2) w 1 + 2 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 ) 2 8 + + q 2 l − 1 ( q 2 l − 1)( q 2 − 1) 8 +(1.6)+(2.2) + ( q 2 l − 1) q 2 l − 1 2 + q l ( q l − 1 +1)( q 2 l − 1) 2 +(3.2)+(4.1) q l ( q l − 1 − 1)( q 2 l − 1) 2 + ( q 2 l − 1)( q 2 l − 2 − 1) q 2( q − 1) (1.4)+(3.1) w 1 + 3 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 ) 2 + q l ( q l − 1 − 1)( q 2 l − 1)( q − 1) 4 (1.5) w 1 + 4 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 )( q − 3) 16 + q 2 l − 1 ( q 2 l − 1)( q − 1) 2 16 T able 9 : W eigh ts and n um b er of co dew ords for q o dd W e no w discus s the case q ev en. Here Q(2 l , q ) has a nuc leus N . 1. Let PG(2 l − 2 , q ) b e an (2 l − 2)- dimensional space in tersecting Q(2 l , q ) in a non- singular (2 l − 2)-dimensional parab olic quadric Q(2 l − 2 , q ). If PG ( 2 l − 2 , q ) is non-n uclear, t hen PG( 2 l − 2 , q ) lies in one tangen t h yp erplane, the h yp erplane h PG(2 l − 2 , q ) , N i , in q / 2 h yp erplanes in tersecting Q( 2 l, q ) in a non- s ingular h yp er- b olic quadric Q + (2 l − 1 , q ), and in q / 2 h yp erplanes inte rsecting Q(2 l, q ) in a non- singular elliptic quadric Q − (2 l − 1 , q ). If PG(2 l − 2 , q ) is nucle ar, then PG(2 l − 2 , q ) lies in q + 1 t a ngen t hyperplanes to Q(2 l , q ). 2. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric P Q + (2 l − 3 , q ), then PG(2 l − 2 , q ) lies in the tangen t h yp erplane to Q(2 l , q ) in P , and in q hyperplanes in tersecting Q(2 l , q ) in non-singular hy p erbolic quadrics Q + (2 l − 1 , q ). 3. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric P Q − (2 l − 3 , q ), then PG(2 l − 2 , q ) lies in the tangen t h yp erplane to Q(2 l , q ) in P , and in q hy p erplanes interse cting Q(2 l , q ) in no n- sin gular elliptic quadrics Q − (2 l − 1 , q ). 4. Let PG(2 l − 2 , q ) b e an (2 l − 2 )-dimens ional space inters ecting Q(2 l , q ) in a singular quadric L Q(2 l − 4 , q ), then PG(2 l − 2 , q ) lies in the ta ng e n t hyperplanes to Q(2 l , q ) in the q + 1 p oin ts P of L . In T able 7, w e denoted the differen t po s sibilities for the in tersection o f Q(2 l , q ) with the union of tw o hyperplanes, a nd in T able 8, the corresp onding sizes for the in tersections. W e no w presen t in T able 10 the n umber of co dew ords having the corresp onding w eights. 15 W eigh t Num b er of co dew ords (1.1) w 1 = q 2 l − 1 − q 2 l − 2 − 2 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 2 )( q − 1) 8 (1.3)+(2.1) w 1 + q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 ) 2 + q l ( q l − 1 +1)( q 2 l − 1)( q − 1) 4 (1.2)+(1.6) w 1 + 2 q l − 1 ( q 2 l − 1) q 2 l ( q − 1 ) 4 + q 2 l − 1 ( q 2 l − 1) 2 + +(4.1) q ( q 2 l − 2 − 1)( q 2 l − 1) 2( q − 1) + +(2.2)+(3.2) q l ( q l − 1 +1)( q 2 l − 1) 2 + q l ( q l − 1 − 1)( q 2 l − 1) 2 (1.4)+(3.1) w 1 + 3 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 ) 2 + q l ( q l − 1 − 1)( q 2 l − 1)( q − 1) 4 (1.5) w 1 + 4 q l − 1 ( q 2 l − 1) q 2 l − 1 ( q − 1 )( q − 2) 8 T able 1 0: W eigh ts and num b er of co dew ords for q ev en Theorem 4.4 L et X b e a no n -de g e ner ate p ar ab olic quadric in PG(2 l , q ) w h e r e l ≥ 1 . A l l the weights w i of the c o de C 2 ( X ) de fi ne d on X ar e divisible b y q l − 1 . Pro of: It is analogous t o the one of Theorem 4.3. References [1] A. Cossiden te and L. Storme, Caps on elliptic quadrics, Finite Fields Appl. 1, (19 95), 412-420 . [2] F. A. B. Edouk ou, Co des correcteurs d’erreurs construits ` a partir des v ari´ et´ es alg ´ ebriques, Ph. D. Thesis, Unive rsit ´ e de la M ´ editerran ´ ee (Aix-Marseille I I), F rance, 2007. [3] F. A. B. Edouk o u, Co des defined b y forms o f degree 2 on quadric surfaces, IEEE T ransactions o n Information Theory , V olume 54, Issue 2, (2008), 860- 864. [4] J.W.P . Hirsc hfeld and J.A. Thas, Gener al Galois Ge ometries . Oxford Mathematical Monographs. Oxford Univers it y Press, 1991. [5] N. M. Katz, On a Theorem o f Ax, American J. of Mathematics, V ol. 93, N0.2, (1 971), 485-499 . Address of the authors: CNRS, Institut de Math ´ ematiques de Lumin y , Lumin y Case 90 7, 13 288 Marseille Cedex 9, F rance. F. A. B. Edoukou: edouk ou@iml.univ-mrs.fr, F. Ro dier: rodier@iml.univ-mrs.fr, h ttp://iml.univ-mrs.fr/ ∼ ro dier/ Departmen t of pure mathematics and computer algebra, Ghen t Univ ersit y , K r ijgslaan 281-S22, 9000 Ghen t, Belgium. A. Hallez: athallez@cage.ugent.be, L. Storme: ls@cage.ugent.be, h ttp://cage.ugen t.b e/ ∼ ls 16